text:
Introduction

This vignette documents the use of the MANOVA.RM package for the analysis of semi-parametric repeated measures designs and multivariate data. The package consists of three parts - one for repeated measurements, one for multivariate data and one for the combination of both - which will be explained in detail below. All functions calculate the Wald-type statistic (WTS) as well as the ANOVA-type statistic (ATS) for repeated measures and a modification thereof (MATS) for multivariate data based on means. These test statistics can be used for arbitrary semi-parametric designs, even with unequal covariance matrices among groups and small sample sizes. For a short overview of the test statistics and corresponding references see the next section. Furthermore, different resampling approaches are provided in order to improve the small sample behavior of the test statistics. The WTS requires non-singular covariance matrices. If the covariance matrix is singular, a warning is returned.

For detailed explanations and examples, see also Friedrich, Konietschke and Pauly (2020).

Theoretical background

We consider the following model $X_{ik} = \mu_i + \varepsilon_{ik}$ for observation vectors from group $$i$$ and individual $$k$$. Here, $$\mu_i$$ denotes the group mean which we wish to infer and $$\varepsilon_{ik}$$ are the common error terms with expectation 0 and existing (co-)variances. We do not need to assume normally distributed errors for these methods and the covariances may be unequal between groups. Null hypotheses are formulated via contrast matrices, i.e., as $$H \mu = 0$$, where the rows of $$H$$ sum to zero. Then the Wald-type statistic (WTS) is a quadratic form in the estimated mean vector involving the Moore-Penrose inverse of the (transformed) empirical covariance matrix. Under the null hypothesis the WTS is asymptotically $$\chi^2$$ distributed with rank(H) degrees of freedom. However, this approximation is only valid for large sample sizes, see e.g. Brunner (2001). We therefore recommend to use a resampling approach as proposed by Konietschke et al. (2015) or Friedrich et al. (2017).

Since the WTS requires non-singular covariance matrices, two other test statistics have been proposed: the ATS for repeated measures designs (Brunner, 2001) and the MATS for multivariate data (Friedrich and Pauly, 2018). The ATS is a scaled quadratic form in the mean vector, which is usually approximated by an F-distribution with estimated degrees of freedom. This approach is also implemented in the SAS PROC Mixed procedure. However, it is rather conservative for small sample sizes and does not provide an asymptotic level $$\alpha$$ test.

The MATS for multivariate data has the advantage of being invariant under scale transformations of the data, an important feature for multivariate data. Since its asymptotic distribution involves unknown parameters, we again propose to use bootstrap approaches instead (Friedrich and Pauly, 2018).

Difference between RM and MANOVA design

• In a repeated measures design (RM) the same outcome is observed at different occasions, e.g., different time points or different parts of the body like left/right hemisphere of the brain. Hypotheses about the sub-plot or within-subject factors are possible.
• In a MANOVA design, d-dimensional outcome vectors are observed for every subject. These outcomes are potentially measured on different scales (e.g., kg, m, heart rate, …) and hypotheses are formulated on the whole outcome vectors. The only difference between the functions MANOVA and MANOVA.wide is the format of the data: For MANOVA data need to be in long format (one row per measurement), whereas MANOVA.wide is for data in wide format (one row per individual).

Note that a combination of both types of data, i.e., multivariate longitudinal data, can now also be analyzed with MANOVA.RM, see below.

The RM function

The RM function calculates the WTS and ATS in a repeated measures design with an arbitrary number of crossed whole-plot (between-subject) and sub-plot (within-subject) factors. The resampling methods provided are a permutation procedure, a parametric bootstrap approach and a wild bootstrap using Rademacher weights. The permutation procedure provides no valid approach for the ATS and is thus not implemented.

Data Example 1 (One between-subject and two within-subject factors)

For illustration purposes, we consider the data set o2cons, which is included in MANOVA.RM.

library(MANOVA.RM)
data(o2cons)

The data set contains measurements on the oxygen consumption of leukocytes in the presence and absence of inactivated staphylococci at three consecutive time points. More details on the study and the statistical model can be found in Friedrich et al. (2017). Due to the study design, both time and staphylococci are within-subject factors while the treatment (Verum vs. Placebo) is a between-subject factor.

head(o2cons)

##     O2 Staphylococci Time Group Subject
## 1 1.48             1    6     P       1
## 2 2.81             1   12     P       1
## 3 3.56             1   18     P       1
## 4 1.04             0    6     P       2
## 5 2.07             0   12     P       2
## 6 2.81             0   18     P       2

We will now analyze this data using the RM function. The RM function takes as arguments:

• formula: A formula consisting of the outcome variable on the left hand side of a ~ operator and the factor variables of interest on the right hand side. An interaction term must be specified. The time variable must be the last factor in the formula.
• data: A data.frame containing the variables in formula.
• subject: The column name of the subjects variable in the data frame.
• no.subf: The number of within-subject factors, default is 1.
• iter: The number of iterations for the resampling approach. Default value is 10,000.
• alpha: The significance level, default is 0.05.
• resampling: The resampling method, one of 'Perm', 'paramBS' or 'WildBS'. Default is set to 'Perm'.
• CPU: The number of cores used for parallel computing. If omitted, cores are detected automatically.
• seed: A random seed for the resampling procedure. If omitted, no reproducible seed is set.
• CI.method: The method for calculating the quantiles used for the confidence intervals, either 't-quantile' (the default) or 'resampling' (based on quantile of the resampled WTS).
• dec: The number of decimals the results should be rounded to. Default is 3.
model1 <- RM(O2 ~ Group * Staphylococci * Time, data = o2cons,
subject = "Subject", no.subf = 2, iter = 1000,
resampling = "Perm", CPU = 1, seed = 1234)
summary(model1)

## Call:
## O2 ~ Group * Staphylococci * Time
## A repeated measures analysis with 2 within-subject factor(s) ( Staphylococci,Time ) and 1 between-subject factor(s).
##
## Descriptive:
##    Group Staphylococci Time  n Means Lower 95 % CI Upper 95 % CI
## 1      P             0    6 12 1.322         1.150         1.493
## 5      P             0   12 12 2.430         2.196         2.664
## 9      P             0   18 12 3.425         3.123         3.727
## 3      P             1    6 12 1.618         1.479         1.758
## 7      P             1   12 12 2.434         2.164         2.704
## 11     P             1   18 12 3.527         3.273         3.781
## 2      V             0    6 12 1.394         1.201         1.588
## 6      V             0   12 12 2.570         2.355         2.785
## 10     V             0   18 12 3.677         3.374         3.979
## 4      V             1    6 12 1.656         1.471         1.840
## 8      V             1   12 12 2.799         2.500         3.098
## 12     V             1   18 12 4.029         3.802         4.257
##
## Wald-Type Statistic (WTS):
##                          Test statistic df  p-value
## Group                    "11.167"       "1" "0.001"
## Staphylococci            "20.401"       "1" "<0.001"
## Group:Staphylococci      "2.554"        "1" "0.11"
## Time                     "4113.057"     "2" "<0.001"
## Group:Time               "24.105"       "2" "<0.001"
## Staphylococci:Time       "4.334"        "2" "0.115"
## Group:Staphylococci:Time "4.303"        "2" "0.116"
##
## ANOVA-Type Statistic (ATS):
##                          Test statistic df1     df2      p-value
## Group                    "11.167"       "1"     "57.959" "0.001"
## Staphylococci            "20.401"       "1"     "Inf"    "<0.001"
## Group:Staphylococci      "2.554"        "1"     "Inf"    "0.11"
## Time                     "960.208"      "1.524" "Inf"    "<0.001"
## Group:Time               "5.393"        "1.524" "Inf"    "0.009"
## Staphylococci:Time       "2.366"        "1.983" "Inf"    "0.094"
## Group:Staphylococci:Time "2.147"        "1.983" "Inf"    "0.117"
##
## p-values resampling:
##                          Perm (WTS)
## Group                    "0.005"
## Staphylococci            "<0.001"
## Group:Staphylococci      "0.129"
## Time                     "<0.001"
## Group:Time               "0.001"
## Staphylococci:Time       "0.17"
## Group:Staphylococci:Time "0.153"

The output consists of four parts: model1$Descriptive gives an overview of the descriptive statistics: The number of observations, mean and confidence intervals (based on either quantiles of the t-distribution or the resampling-based quantiles) are displayed for each factor level combination. model1$WTS contains the results for the Wald-type test: The test statistic, degree of freedom and p-values based on the asymptotic $$\chi^2$$ distribution are displayed. Note that the $$\chi^2$$ approximation is very liberal for small sample sizes. model1$ATS contains the corresponding results based on the ATS. This test statistic tends to rather conservative decisions in case of small sample sizes and is even asymptotically only an approximation, thus not providing an asymptotic level $$\alpha$$ test. Finally, model1$resampling contains the p-values based on the chosen resampling approach. For the ATS, the permutation procedure cannot be applied. Due to the above mentioned issues for small sample sizes, the resampling procedure is recommended.

Data Example 2 (Two within-subject and two between-subject factors)

We consider the data set EEG from the MANOVA.RM package: At the Department of Neurology, University Clinic of Salzburg, 160 patients were diagnosed with either Alzheimer's Disease (AD), mild cognitive impairments (MCI), or subjective cognitive complaints without clinically significant deficits (SCC), based on neuropsychological diagnostics (Bathke et al.(2018)). This data set contains z-scores for brain rate and Hjorth complexity, each measured at frontal, temporal and central electrode positions and averaged across hemispheres. In addition to standardization, complexity values were multiplied by -1 in order to make them more easily comparable to brain rate values: For brain rate we know that the values decrease with age and pathology, while Hjorth complexity values are known to increase with age and pathology. The three between-subject factors considered were sex (men vs. women), diagnosis (AD vs. MCI vs. SCC), and age ($$< 70$$ vs. $$>= 70$$ years). Additionally, the within-subject factors region (frontal, temporal, central) and feature (brain rate, complexity) structure the response vector.

data(EEG)
EEG_model <- RM(resp ~ sex * diagnosis * feature * region,
data = EEG, subject = "id", no.subf = 2, resampling = "WildBS",
iter = 1000,  alpha = 0.01, CPU = 1, seed = 987)
summary(EEG_model)

## Call:
## resp ~ sex * diagnosis * feature * region
## A repeated measures analysis with 2 within-subject factor(s) ( feature,region ) and 2 between-subject factor(s).
##
## Descriptive:
##    sex diagnosis    feature   region  n  Means Lower 99 % CI Upper 99 % CI
## 1    M        AD  brainrate  central 12 -1.010        -4.881         2.861
## 13   M        AD  brainrate  frontal 12 -1.007        -4.991         2.977
## 25   M        AD  brainrate temporal 12 -0.987        -4.493         2.519
## 7    M        AD complexity  central 12 -1.488       -10.053         7.077
## 19   M        AD complexity  frontal 12 -1.086        -6.906         4.735
## 31   M        AD complexity temporal 12 -1.320        -7.203         4.562
## 3    M       MCI  brainrate  central 27 -0.447        -1.591         0.696
## 15   M       MCI  brainrate  frontal 27 -0.464        -1.646         0.719
## 27   M       MCI  brainrate temporal 27 -0.506        -1.584         0.572
## 9    M       MCI complexity  central 27 -0.257        -1.139         0.625
## 21   M       MCI complexity  frontal 27 -0.459        -1.997         1.079
## 33   M       MCI complexity temporal 27 -0.490        -1.796         0.816
## 5    M       SCC  brainrate  central 20  0.459        -0.414         1.332
## 17   M       SCC  brainrate  frontal 20  0.243        -0.670         1.156
## 29   M       SCC  brainrate temporal 20  0.409        -1.210         2.028
## 11   M       SCC complexity  central 20  0.349        -0.070         0.767
## 23   M       SCC complexity  frontal 20  0.095        -1.037         1.227
## 35   M       SCC complexity temporal 20  0.314        -0.598         1.226
## 2    W        AD  brainrate  central 24 -0.294        -1.978         1.391
## 14   W        AD  brainrate  frontal 24 -0.159        -1.813         1.495
## 26   W        AD  brainrate temporal 24 -0.285        -1.776         1.206
## 8    W        AD complexity  central 24 -0.128        -1.372         1.116
## 20   W        AD complexity  frontal 24  0.026        -1.212         1.264
## 32   W        AD complexity temporal 24 -0.194        -1.670         1.283
## 4    W       MCI  brainrate  central 30 -0.106        -1.076         0.863
## 16   W       MCI  brainrate  frontal 30 -0.074        -1.032         0.885
## 28   W       MCI  brainrate temporal 30 -0.069        -1.064         0.925
## 10   W       MCI complexity  central 30  0.094        -0.464         0.652
## 22   W       MCI complexity  frontal 30  0.131        -0.768         1.031
## 34   W       MCI complexity temporal 30  0.121        -0.652         0.895
## 6    W       SCC  brainrate  central 47  0.537        -0.049         1.124
## 18   W       SCC  brainrate  frontal 47  0.548        -0.062         1.159
## 30   W       SCC  brainrate temporal 47  0.559        -0.015         1.133
## 12   W       SCC complexity  central 47  0.384         0.110         0.659
## 24   W       SCC complexity  frontal 47  0.403        -0.038         0.845
## 36   W       SCC complexity temporal 47  0.506         0.132         0.880
##
## Wald-Type Statistic (WTS):
##                              Test statistic df  p-value
## sex                          "9.973"        "1" "0.002"
## diagnosis                    "42.383"       "2" "<0.001"
## sex:diagnosis                "3.777"        "2" "0.151"
## feature                      "0.086"        "1" "0.769"
## sex:feature                  "2.167"        "1" "0.141"
## diagnosis:feature            "5.317"        "2" "0.07"
## sex:diagnosis:feature        "1.735"        "2" "0.42"
## region                       "0.07"         "2" "0.966"
## sex:region                   "0.876"        "2" "0.645"
## diagnosis:region             "6.121"        "4" "0.19"
## sex:diagnosis:region         "1.532"        "4" "0.821"
## feature:region               "0.652"        "2" "0.722"
## sex:feature:region           "0.423"        "2" "0.81"
## diagnosis:feature:region     "7.142"        "4" "0.129"
## sex:diagnosis:feature:region "2.274"        "4" "0.686"
##
## ANOVA-Type Statistic (ATS):
##                              Test statistic df1     df2      p-value
## sex                          "9.973"        "1"     "36.497" "0.003"
## diagnosis                    "13.124"       "1.343" "36.497" "<0.001"
## sex:diagnosis                "1.904"        "1.343" "36.497" "0.174"
## feature                      "0.086"        "1"     "Inf"    "0.769"
## sex:feature                  "2.167"        "1"     "Inf"    "0.141"
## diagnosis:feature            "1.437"        "1.562" "Inf"    "0.238"
## sex:diagnosis:feature        "1.031"        "1.562" "Inf"    "0.342"
## region                       "0.018"        "1.611" "Inf"    "0.965"
## sex:region                   "0.371"        "1.611" "Inf"    "0.644"
## diagnosis:region             "1.091"        "2.046" "Inf"    "0.337"
## sex:diagnosis:region         "0.376"        "2.046" "Inf"    "0.691"
## feature:region               "0.126"        "1.421" "Inf"    "0.81"
## sex:feature:region           "0.077"        "1.421" "Inf"    "0.864"
## diagnosis:feature:region     "0.829"        "1.624" "Inf"    "0.415"
## sex:diagnosis:feature:region "0.611"        "1.624" "Inf"    "0.51"
##
## p-values resampling:
##                              WildBS (WTS) WildBS (ATS)
## sex                          "0.001"      "0.001"
## diagnosis                    "<0.001"     "<0.001"
## sex:diagnosis                "0.127"      "0.129"
## feature                      "0.797"      "0.797"
## sex:feature                  "0.157"      "0.157"
## diagnosis:feature            "0.054"      "0.257"
## sex:diagnosis:feature        "0.464"      "0.363"
## region                       "0.969"      "0.981"
## sex:region                   "0.688"      "0.7"
## diagnosis:region             "0.168"      "0.332"
## sex:diagnosis:region         "0.865"      "0.82"
## feature:region               "0.817"      "0.933"
## sex:feature:region           "0.871"      "0.944"
## diagnosis:feature:region     "0.104"      "0.502"
## sex:diagnosis:feature:region "0.738"      "0.682"

We find significant effects at level $$\alpha = 0.01$$ of the between-subject factors sex and diagnosis, while none of the within-subject factors or interactions become significant.

Plotting

The RM() function is equipped with a plotting option, displaying the calculated means along with $$(1-\alpha)$$ confidence intervals based on a $$t$$-approximation. The plot function takes an RM object as an argument. In addition, the factor of interest may be specified. If this argument is omitted in a two- or higher-way layout, the user is asked to specify the factor for plotting. Furthermore, additional graphical parameters can be used to customize the plots. The optional argument legendpos specifies the position of the legend in higher-way layouts, while the argument gap (default 0.1) specifies the distance between the error bars.

plot(EEG_model, factor = "sex", main = "Effect of sex on EEG values")

plot(EEG_model, factor = "sex:diagnosis", legendpos = "topleft", col = c(4, 2))

plot(EEG_model, factor = "sex:diagnosis:feature", legendpos = "center", gap = 0.05)

The MANOVA function

The MANOVA function calculates the WTS for multivariate data in a design with crossed or nested factors. Additionally, a modified ANOVA-type statistic (MATS) is calculated which has the additional advantage of being applicable to designs involving singular covariance matrices and is invariant under scale transformations of the data, see Friedrich and Pauly (2018) for details. The resampling methods provided are a parametric bootstrap approach and a wild bootstrap using Rademacher weights. Note that only balanced nested designs (i.e., the same number of factor levels $$b$$ for each level of the factor $$A$$) with up to three factors are implemented. Designs involving both crossed and nested factors are not implemented. Data must be provided in long format (for wide format, see MANOVA.wide below).

Data Example MANOVA (two crossed factors)

We again consider the data set EEG from the MANOVA.RM package, but now we ignore the within-subject factors. Therefore, we are now in a multivariate setting with 6 measurements per patient and three crossed factors sex, age and diagnosis. Due to the small number of subjects in some groups (e.g., only 2 male patients aged $$<$$ 70 were diagnosed with AD) we restrict our analyses to two factors at a time. The analysis of this example is shown below.

The MANOVA function takes as arguments:

• formula: A formula consisting of the outcome variable on the left hand side of a ~ operator and the factor variables of interest on the right hand side.
• data: A data.frame containing the variables in formula.
• subject: The column name of the subjects variable in the data frame.
• iter: The number of iterations for the resampling procedure. Default value is 10,000.
• alpha: The significance level, default is 0.05.
• resampling: The resampling method, one of 'paramBS' and 'WildBS'. Default is set to 'paramBS'.
• CPU: The number of cores used for parallel computing. If omitted, cores are detected automatically.
• seed: A random seed for the resampling procedure. If omitted, no reproducible seed is set.
• nested.levels.unique: For nested designs only: A logical specifying whether the levels of the nested factor(s) are labeled uniquely or not. Default is FALSE, i.e., the levels of the nested factor are the same for each level of the main factor. For an example and more explanations see the GFD package and the corresponding vignette.
• dec: The number of decimals the results should be rounded to. Default is 3.
data(EEG)
EEG_MANOVA <- MANOVA(resp ~ sex * diagnosis,
data = EEG, subject = "id", resampling = "paramBS",
iter = 1000,  alpha = 0.01, CPU = 1, seed = 987)
summary(EEG_MANOVA)

## Call:
## resp ~ sex * diagnosis
##
## Descriptive:
##   sex diagnosis  n Mean 1 Mean 2 Mean 3 Mean 4 Mean 5 Mean 6
## 1   M        AD 12 -0.987 -1.007 -1.010 -1.320 -1.086 -1.488
## 3   M       MCI 27 -0.506 -0.464 -0.447 -0.490 -0.459 -0.257
## 5   M       SCC 20  0.409  0.243  0.459  0.314  0.095  0.349
## 2   W        AD 24 -0.285 -0.159 -0.294 -0.194  0.026 -0.128
## 4   W       MCI 30 -0.069 -0.074 -0.106  0.121  0.131  0.094
## 6   W       SCC 47  0.559  0.548  0.537  0.506  0.403  0.384
##
## Wald-Type Statistic (WTS):
##               Test statistic df   p-value
## sex           "12.604"       "6"  "0.05"
## diagnosis     "55.158"       "12" "<0.001"
## sex:diagnosis "9.79"         "12" "0.634"
##
## modified ANOVA-Type Statistic (MATS):
##               Test statistic
## sex                   45.263
## diagnosis            194.165
## sex:diagnosis         18.401
##
## p-values resampling:
##               paramBS (WTS) paramBS (MATS)
## sex           "0.124"       "0.003"
## diagnosis     "<0.001"      "<0.001"
## sex:diagnosis "0.748"       "0.223"

The output consists of several parts: First, some descriptive statistics of the data set are displayed, namely the sample size and mean for each factor level combination and each dimension. (Dimensions occur in the same order as in the original data set. For a labeled output, use MANOVA.wide().) In this example, Mean 1 to Mean 3 correspond to the brainrate (temporal, frontal, central) while Mean 4–6 correspond to complexity. Second, the results based on the WTS are displayed. For each factor, the test statistic, degree of freedom and p-value is given. For the MATS, only the value of the test statistic is given, since inference is here based on the resampling procedure only. The resampling-based p-values are finally displayed for both test statistics.

The MANOVA.wide function

The MANOVA.wide function is used for data provided in wide format, i.e., with one row per unit. Input and output are almost identical to the MANOVA function, except that no subject variable needs to be specified. The formula now consists of the matrix of outcome variables (bound together via cbind()) on the left hand side of the ~ operator and the factors of interest on the right. For an example we use a data set on producing plastic film from Krzanowski (1998, p. 381), see also summary.manova:

tear <- c(6.5, 6.2, 5.8, 6.5, 6.5, 6.9, 7.2, 6.9, 6.1, 6.3,
6.7, 6.6, 7.2, 7.1, 6.8, 7.1, 7.0, 7.2, 7.5, 7.6)
gloss <- c(9.5, 9.9, 9.6, 9.6, 9.2, 9.1, 10.0, 9.9, 9.5, 9.4,
9.1, 9.3, 8.3, 8.4, 8.5, 9.2, 8.8, 9.7, 10.1, 9.2)
opacity <- c(4.4, 6.4, 3.0, 4.1, 0.8, 5.7, 2.0, 3.9, 1.9, 5.7,
2.8, 4.1, 3.8, 1.6, 3.4, 8.4, 5.2, 6.9, 2.7, 1.9)
rate     <- gl(2,10, labels = c("Low", "High"))
additive <- gl(2, 5, length = 20, labels = c("Low", "High"))

example <- data.frame(tear, gloss, opacity, rate, additive)
fit <- MANOVA.wide(cbind(tear, gloss, opacity) ~ rate * additive, data = example, iter = 1000, CPU = 1)
summary(fit)

## Call:
## cbind(tear, gloss, opacity) ~ rate * additive
##
## Descriptive:
##   rate additive n tear  gloss  opacity
## 1  Low      Low 5 6.30   9.56     3.74
## 2  Low     High 5 6.68   9.58     3.84
## 3 High      Low 5 6.88   8.72     3.14
## 4 High     High 5 7.28   9.40     5.02
##
## Wald-Type Statistic (WTS):
##               Test statistic df  p-value
## rate          "25.9"         "3" "<0.001"
##
## modified ANOVA-Type Statistic (MATS):
##               Test statistic
## rate                  23.808
##
## p-values resampling:
##               paramBS (WTS) paramBS (MATS)
## rate          "0.004"       "<0.001"

Confidence regions

A function for calculating and plotting of confidence regions is available for MANOVA objects. Details on the methods can be found in Friedrich and Pauly (2018).

Confidence regions

Confidence regions can be calculated using the conf.reg function. Note that confidence regions can only be plotted in designs with 2 dimensions. The conf.reg function takes as arguments:

• object: A MANOVA object calculated via MANOVA() or MANOVA.wide().
• nullhypo: In designs involving more than one factor, it is necessary to specify the null hypothesis, i.e., the contrast of interest.

As an example, we consider the data set water from the package HSAUR3. The data set contains measurements of mortality and drinking water hardness for 61 cities in England and Wales. Suppose we want to analyse whether these measurements differ between northern and southern towns. Since the data set is in wide format, we need to use the MANOVA.wide function.

if (requireNamespace("HSAUR3", quietly = TRUE)) {
library(HSAUR3)
data(water)
test <- MANOVA.wide(cbind(mortality, hardness) ~ location, data = water, iter = 1000, resampling = "paramBS", CPU = 1, seed = 123)
summary(test)
cr <- conf.reg(test)
cr
}

## Loading required package: tools

## Call:
## cbind(mortality, hardness) ~ location
##
## Descriptive:
##   location  n mortality  hardness
## 1    North 35  1633.600    30.400
## 2    South 26  1376.808    69.769
##
## Wald-Type Statistic (WTS):
##          Test statistic df  p-value
## location "51.584"       "2" "<0.001"
##
## modified ANOVA-Type Statistic (MATS):
##          Test statistic
## location         69.882
##
## p-values resampling:
##          paramBS (WTS) paramBS (MATS)
## location "<0.001"      "<0.001"

## Center:
##         [,1]
## [1,] 256.792
## [2,] -39.369
##
## Scale:
## [1] 90.70294 22.86942
##
## Eigenvectors:
##      [,1] [,2]
## [1,]   -1    0
## [2,]    0   -1

The output consists of the necessary parameters specifying the ellipsoid: the center, the eigenvectors which determine the axes of the ellipsoid as well as the scaling factors for the eigenvectors, which are calculated based on the eigenvalues, the bootstrap quantile and the total sample size. For more information on the construction of the confidence ellipses see Friedrich and Pauly (2018). For observations with two dimensions, the confidence ellipse can be plotted using the generic plot function. The usual plotting parameters can be used to customize the plots.

plot(cr, col = 2, lty = 2, xlab = "Difference in mortality", ylab ="Difference in water hardness")

Post-hoc comparisons

Multivariate post-hoc comparisons and simultaneous confidence intervals for contrasts

Calculation of simultaneous confidence intervals and multivariate p-values for contrasts of the mean vector is based on the sum statistic, see Friedrich and Pauly (2018) for details. Note that the confidence intervals and p-values returned are simultaneous, i.e., they maintain the given alpha-level. Confidence intervals are calculated based on summary effects, i.e., averaging over all dimensions, whereas the returned p-values are multivariate. If the original model contains more than one factor, the corresponding contrasts are calculated for all combinations of factor levels. If this is not desired, the parameter interaction may be set to FALSE and the factor of interest specified. This has the same effect as fitting the model with only the respective factor and then calculating the post-hoc tests based on this model. The function simCI takes the following arguments:

• object: A MANOVA object calculated via MANOVA() or MANOVA.wide().
• contrast: The contrast of interest, can either be “pairwise” or “user-defined”.
• contmat: For user-defined contrasts, the contrast matrix must be specified here. Note that its rows must sum to zero.
• type: Pairwise contrasts are calculated based on the contrMat function in package multcomp, see also the corresponding help page. The type of the pairwise comparison must be specified here.
• base: an integer specifying which group is considered the baseline group for Dunnett contrasts, see also the documentation of contrMat() from the multcomp-package.
• interaction: Logical. If interaction = FALSE in models with more than one factor, the factor of interest for the post-hoc analysis must be specified. Default is TRUE, which means post-hoc tests are performed for all factor level combinations.
• factor: Only needed if interaction = FALSE. Specifies the factor for which post-hoc analysis are requested.
• silent: Set to TRUE to suppress output.

As an example, we consider the EEG_MANOVA example from above:

# pairwise comparison using Tukey contrasts
simCI(EEG_MANOVA, contrast = "pairwise", type = "Tukey")

##
##  #------ Call -----#
##
##  - Contrast:  Tukey
##  - Confidence level: 99 %
##
##  #------Multivariate post-hoc comparisons: p-values -----#
##
##         contrast p.value
## 1   M MCI - M AD   0.961
## 2   M SCC - M AD   0.548
## 4   W MCI - M AD   0.775
## 5   W SCC - M AD   0.417
## 6  M SCC - M MCI   0.368
## 7   W AD - M MCI   0.995
## 8  W MCI - M MCI   0.843
## 9  W SCC - M MCI   0.111
## 10  W AD - M SCC   0.845
## 11 W MCI - M SCC   0.938
## 12 W SCC - M SCC   0.989
## 13  W MCI - W AD   1.000
## 14  W SCC - W AD   0.526
## 15 W SCC - W MCI   0.563
##
##  #-----------Confidence intervals for summary effects-------------#
##
##               Estimate      Lower     Upper
## M MCI - M AD     4.275 -16.181707 24.731707
## M SCC - M AD     8.767 -11.126510 28.660510
## W MCI - M AD     6.995 -12.978374 26.968374
## W SCC - M AD     9.835  -9.799574 29.469574
## M SCC - M MCI    4.492  -4.040216 13.024216
## W AD - M MCI     1.589  -8.907565 12.085565
## W MCI - M MCI    2.720  -5.996803 11.436803
## W SCC - M MCI    5.560  -2.349709 13.469709
## W AD - M SCC    -2.903 -12.254617  6.448617
## W MCI - M SCC   -1.772  -9.069776  5.525776
## W SCC - M SCC    1.068  -5.243764  7.379764
## W MCI - W AD     1.131  -8.389330 10.651330
## W SCC - W AD     3.971  -4.816350 12.758350
## W SCC - W MCI    2.840  -3.719140  9.399140

# the same but with Dunnett contrasts using group 2 as baseline
#simCI(EEG_MANOVA, contrast = "pairwise", type = "Dunnett", base = 2, interaction = FALSE, factor = "diagnosis")
# a one-way layout using MANOVA.wide():
oneway <- MANOVA.wide(cbind(brainrate_temporal, brainrate_central) ~ diagnosis, data = EEGwide,
iter = 1000, CPU = 1)
# and a user-defined contrast matrix
H <- as.matrix(cbind(rep(1, 5), -1*Matrix::Diagonal(5)))
# user-specified comparison
simCI(oneway, contrast = "user-defined", contmat = H)

##
##  #------ Call -----#
##
##  - Contrast:  user-defined
##  - Confidence level: 95 %
##
##  #------Multivariate post-hoc comparisons: p-values -----#
##
## [1] 1.000 0.720 0.701 0.013 0.010
##
##  #-----------Confidence intervals for summary effects-------------#
##
##   Estimate     Lower      Upper
## 1    0.013 -1.025882  1.0518818
## 2   -0.243 -1.067879  0.5818789
## 3   -0.251 -1.076162  0.5741618
## 4   -1.033 -1.829751 -0.2362493
## 5   -1.033 -1.808464 -0.2575362

Univariate comparisons

If the global null hypothesis is rejected, it may be of interest to infer the univariate outcomes that caused the rejection. To answer this question, one can simply calculate the univariate p-values and adjust them accordingly for multiple testing, e.g., using Bonferroni-correction. An example is given below. We consider a one-way layout of the EEG data with influencing factor sex and all 6 outcome variables:

model_sex <- MANOVA.wide(cbind(brainrate_temporal, brainrate_central, brainrate_frontal,
complexity_temporal, complexity_central, complexity_frontal) ~ sex,
data = EEGwide,
iter = 1000, seed = 987, CPU = 1)
summary(model_sex)

## Call:
## cbind(brainrate_temporal, brainrate_central, brainrate_frontal,
##     complexity_temporal, complexity_central, complexity_frontal) ~
##     sex
##
## Descriptive:
##   sex   n brainrate_temporal  brainrate_central  brainrate_frontal
## 1   M  59             -0.294             -0.254             -0.335
## 2   W 101              0.172              0.149              0.195
##    complexity_temporal  complexity_central  complexity_frontal
## 1               -0.386              -0.302              -0.399
## 2                0.226               0.176               0.233
##
## Wald-Type Statistic (WTS):
##     Test statistic df  p-value
## sex "15.382"       "6" "0.017"
##
## modified ANOVA-Type Statistic (MATS):
##     Test statistic
## sex         55.725
##
## p-values resampling:
##     paramBS (WTS) paramBS (MATS)
## sex "0.031"       "<0.001"

Since the global hypothesis is rejected at 5% level, we continue with the univariate calculations:

EEG1 <- MANOVA.wide(brainrate_temporal ~ sex, data = EEGwide, iter = 1000, seed = 987, CPU = 1)
EEG2 <- MANOVA.wide(brainrate_central ~ sex, data = EEGwide, iter = 1000, seed = 987, CPU = 1)
EEG3 <- MANOVA.wide(brainrate_frontal ~ sex, data = EEGwide, iter = 1000, seed = 987, CPU = 1)
EEG4 <- MANOVA.wide(complexity_temporal ~ sex, data = EEGwide, iter = 1000, seed = 987, CPU = 1)
EEG5 <- MANOVA.wide(complexity_central ~ sex, data = EEGwide, iter = 1000, seed = 987, CPU = 1)
EEG6 <- MANOVA.wide(complexity_frontal ~ sex, data = EEGwide, iter = 1000, seed = 987, CPU = 1)

Adjust for multiple testing using the parametric bootstrap MATS and Bonferroni adjustment:

p.adjust(c(EEG1$resampling[, 2], EEG2$resampling[, 2], EEG3$resampling[, 2], EEG4$resampling[, 2], EEG5$resampling[, 2], EEG6$resampling[, 2]),
method = "bonferroni")

## [1] 0.036 0.078 0.012 0.012 0.096 0.006

This reveals that the central variables (comparison 2 and 5) do not contribute to the significant difference between male and female patients.

Nested Design

To create a data example for a nested design, we use the curdies data set from the GFD package and extend it by introducing an artificial second outcome variable. In this data set, the levels of the nested factor (site) are named uniquely, i.e., levels 1-3 of factor site belong to “WINTER”, whereas levels 4-6 belong to “SUMMER”. Therefore, nested.levels.unique must be set to TRUE. The code for the analysis using both wide and long format is presented below.

if (requireNamespace("GFD", quietly = TRUE)) {
library(GFD)
data(curdies)
set.seed(123)
curdies$dug2 <- curdies$dugesia + rnorm(36)

# first possibility: MANOVA.wide
fit1 <- MANOVA.wide(cbind(dugesia, dug2) ~ season + season:site, data = curdies, iter = 1000, nested.levels.unique = TRUE, seed = 123, CPU = 1)

# second possibility: MANOVA (long format)
dug <- c(curdies$dugesia, curdies$dug2)
season <- rep(curdies$season, 2) site <- rep(curdies$site, 2)
curd <- data.frame(dug, season, site, subject = rep(1:36, 2))

fit2 <- MANOVA(dug ~ season + season:site, data = curd, subject = "subject", nested.levels.unique = TRUE, seed = 123, iter = 1000,
CPU = 1)

# comparison of results
summary(fit1)
summary(fit2)
}

## Call:
## cbind(dugesia, dug2) ~ season + season:site
##
## Descriptive:
##         season site n dugesia   dug2
## 1 SUMMER          4 6   0.419 -0.050
## 2 SUMMER          5 6   0.229  0.028
## 3 SUMMER          6 6   0.194  0.763
## 4 WINTER          1 6   2.049  2.497
## 5 WINTER          2 6   4.182  4.123
## 6 WINTER          3 6   0.678  0.724
##
## Wald-Type Statistic (WTS):
##             Test statistic df  p-value
## season      "6.999"        "2" "0.03"
## season:site "16.621"       "8" "0.034"
##
## modified ANOVA-Type Statistic (MATS):
##             Test statistic
## season              12.296
## season:site         15.064
##
## p-values resampling:
##             paramBS (WTS) paramBS (MATS)
## season      "0.064"       "0.032"
## season:site "0.275"       "0.216"
## Call:
## dug ~ season + season:site
##
## Descriptive:
##         season site n Mean 1 Mean 2
## 1 SUMMER          4 6  0.419 -0.050
## 2 SUMMER          5 6  0.229  0.028
## 3 SUMMER          6 6  0.194  0.763
## 4 WINTER          1 6  2.049  2.497
## 5 WINTER          2 6  4.182  4.123
## 6 WINTER          3 6  0.678  0.724
##
## Wald-Type Statistic (WTS):
##             Test statistic df  p-value
## season      "6.999"        "2" "0.03"
## season:site "16.621"       "8" "0.034"
##
## modified ANOVA-Type Statistic (MATS):
##             Test statistic
## season              12.296
## season:site         15.064
##
## p-values resampling:
##             paramBS (WTS) paramBS (MATS)
## season      "0.064"       "0.032"
## season:site "0.275"       "0.216"

The multRM() function

The multRM() function provides a combination of the approaches described above. It is suitable for repeated measures designs, in which multiple outcomes have been recorded at each time point. The multRM() function takes as arguments:

• formula: A model formula object. The left hand side contains the matrix of response variables (using cbind()) and the right hand side contains the factor variables of interest.
• data: A data.frame, list or environment containing the variables in formula. Data must be in long format and must not contain missing values.
• subject: The column name of the subjects in the data. NOTE: Subjects within different groups of between-subject factors must have individual labels.
• within: Specifies the within-subject factor(s) in the formula.
• iter: The number of iterations used for calculating the resampled statistic. The default option is 10,000.
• alpha: A number specifying the significance level; the default is 0.05.
• resampling: The resampling method to be used, one of “paramBS” (parametric bootstrap approach) and “WildBS” (wild bootstrap approach with Rademacher weights).
• CPU: The number of cores used for parallel computing. If omitted, cores are detected automatically.
• seed: A random seed for the resampling procedure. If omitted, no reproducible seed is set.
• dec: Number of decimals the results should be rounded to. Default is 3.

As an example, we again use the EEG dataset. This time, imagine we have two outcomes (brainrate and complexity) measured for each of the three regions (within-subject factor). We additionally consider the between-subject factor sex. The tidyr package can be used to transform our original data to this format.

library(tidyr)

##   sex age diagnosis   region  id brainrate complexity
## 1   M   0        AD  central  65 -1.141140 -1.2560418
## 2   M   0        AD  central 155 -2.013742 -2.8668007
## 3   M   0        AD  frontal  65 -1.331298 -2.5610352
## 4   M   0        AD  frontal 155 -1.552207 -0.7810009
## 5   M   0        AD temporal  65 -1.092243 -1.2511514
## 6   M   0        AD temporal 155 -2.055178 -2.8887544

fit <- multRM(cbind(brainrate, complexity) ~ sex * region, data = eeg, subject = "id", within = "region", iter = 200, CPU = 1)
summary(fit)

## Call:
## cbind(brainrate, complexity) ~ sex * region
## A multivariate repeated measures analysis with  1 within-subject factor(s) ( region )and  1 between-subject factor(s).
##
## Descriptive:
##   sex   region   n brainrate  complexity
## 1   M  central  59    -0.254      -0.302
## 2   M  frontal 101    -0.335      -0.399
## 3   M temporal  59    -0.294      -0.386
## 4   W  central 101     0.149       0.176
## 5   W  frontal  59     0.195       0.233
## 6   W temporal 101     0.172       0.226
##
## Wald-Type Statistic (WTS):
##            Test statistic df  p-value
## sex        "12.45"        "2" "0.002"
## region     "0.192"        "4" "0.996"
## sex:region "2.79"         "4" "0.593"
##
## modified ANOVA-Type Statistic (MATS):
##            Test statistic
## sex                54.203
## region              0.048
## sex:region          0.703
##
## p-values resampling:
##            paramBS (WTS) paramBS (MATS)
## sex        "0.005"       "0.005"
## region     "0.995"       "0.985"
## sex:region "0.58"        "0.395"

The output is similar to that of MANOVA() described above.

optional GUI

The MANOVA.RM package is equipped with an optional graphical user interface, which is based on RGtk2. The GUI may be started in R (if RGtk2 is installed) using the command GUI.RM() and GUI.MANOVA() or GUI.MANOVAwide() for repeated measures designs and multivariate data, respectively.

if (requireNamespace("RGtk2", quietly = TRUE)) {
GUI.MANOVA()
}

The user can specify the data location (either directly or via the “load data” button), the formula, the number of iterations for the resampling approach and the significance level. Furthermore, one needs to specify the number of within-subject factors (for the repeated measures design only), the 'subject' variable in the data frame and the resampling method. Additionally, one can specify whether or not headers are included in the data file, and which separator (e.g., ',' for *.csv files) and character symbols are used for decimals in the data file. The GUI for RM also provides a plotting option, which generates a new window for specifying the factors to be plotted (in higher-way layouts) along with a few plotting parameters. For the multRM() function there is no GUI available yet.

References

• Bathke, A. et al. (2018). Testing Mean Differences among Groups: Multivariate and Repeated Measures Analysis with Minimal Assumptions. Multivariate Behavioral Research, 53(3), 348-359, DOI: 10.1080/00273171.2018.1446320.

• Brunner, E. (2001). Asymptotic and approximate analysis of repeated measures designs under heteroscedasticity. Mathematical Statistics with Applications in Biometry.

• Friedrich, S., Brunner, E. and Pauly, M. (2017). Permuting longitudinal data in spite of the dependencies. Journal of Multivariate Analysis, 153, 255-265.

• Friedrich, S., and Pauly, M. (2018). MATS: Inference for potentially singular and heteroscedastic MANOVA. Journal of Multivariate Analysis, 165, 166-179, DOI: 10.1016/j.jmva.2017.12.008.

• Friedrich, S., Konietschke, F., and Pauly, M. (2019). Resampling-Based Analysis of Multivariate Data and Repeated Measures Designs with the R Package MANOVA.RM. The R Journal, 11(2), 380-400.

• Konietschke, F., Bathke, A. C., Harrar, S. W. and Pauly, M. (2015). Parametric and nonparametric bootstrap methods for general MANOVA. Journal of Multivariate Analysis, 140, 291-301.


text:
# special table in latex

I would like to make a table like this in the image please can someone tell how I can do that

    \begin{table}[!h]
\begin{center}
\begin{tabular}{l|p{1.2cm}|p{1.2cm}|p{1.2cm}|p{1.2cm}|p{1.2cm}|p{1.2cm}|p{1.2cm}|p{1.2cm}|p{1.2cm}|p{1.2cm}|p{1.2cm}}
\cline{2-10}
&\multicolumn{3}{|c|} {\bf $N_{min}$ Amis }& \multicolumn{3}{|c|} {\bf $N_{moy}$ Amis  }& \multicolumn{3}{|c|} {\bf $N_{max}$ Amis }\\ %\multicolumn{2}{|p{2.5cm}|} {\bf 750 Pages }& %\multicolumn{2}{|p{2.5cm}|} {\bf 1000 Pages }\\
\cline{2-10}
& \bf AS& \bf A\& A & \bf Pub& \bf AS& \bf A\& A & \bf Pub & \bf AS & \bf A\& A & \bf Pub \\
\hline

\multicolumn{1}{|l|}{$N_{moy}$ Amis} &\bf 25.66& \bf 26.66& \bf 27&\bf 33& \bf 33& \bf 33& \bf 39.66& \bf 39.66& \bf 39.5\\
\hline
\multicolumn{1}{|p{3.5cm}|}{Moy Pages cibles } &\bf 25.66& \bf 62& \bf 100& \bf 33& \bf 71.33& \bf 100& \bf 39.66& \bf 80.33& \bf 100\\

\hline

\end{tabular}
\end{center}

\bf{\caption{Partager Publication dans le réseau 1}}
\end{table}

-
but the code that I propose is right but it is missing a column in the left before Nmoy Min and Nb Pages cibles –  boutheina May 9 '12 at 21:48
I modify the image do you understand what I want? –  boutheina May 9 '12 at 21:58

It is virtually impossible to rela te your image to the posted code, but anyway I offer this, if it is not what you need then please edit the question to make it clearer. Also please note that \bf is an obsolete command (and even when used never takes an argument. It is not \bf{...} but {\bf ...} or better \bfseries.

\documentclass{article}

\begin{document}

\noindent X\dotfill X

\begin{table}[htbp]
\centering
\footnotesize
\bfseries

\hspace*{-2cm}\begin{tabular}{|l|l|*{11}{p{.8cm}|}}
\cline{3-11}
\multicolumn{2}{l|}{}&\multicolumn{3}{|c|} { $N_{min}$ Amis }& \multicolumn{3}{c|} { $N_{moy}$ Amis  }& \multicolumn{3}{c|} { $N_{max}$ Amis }\\
\cline{3-11}
\multicolumn{2}{l|}{}&  AS&  A\&A &  Pub&  AS&  A\&A &  Pub &  AS &  A\&A &  Pub \\
\hline
\raisebox{-5pt}{xxx}&$N_{moy}$ Amis & 25.66&  26.66&  27& 33&  33&  33&  39.66&  39.66&  39.5\\
\cline{2-11}
&Moy Pages cibles & 25.66&  62&  100&  33&  71.33&  100&  39.66&  80.33&  100\\
\hline
\end{tabular}\hspace*{-2cm}
\end{table}

\noindent X\dotfill X

\end{document}

-
the code that you suggest gives the same result of the code that I proposed. But I want that before the two columns entitled N moyAmis Nb pages cibles we add a new cell that contain Num Réseau –  boutheina May 9 '12 at 23:03
the code here fits on the page unlike the code in the question, how was anyone supposed to deduce that you wanted an extra cell with some text that doesn't appear anywhere in the question? I still have no idea where you want that text to appear. I suggest that you take the image from my answer and edit it in a bitmap editor and just add the text you want at the right place. that way we might have a clue, and can update the code to match. –  David Carlisle May 9 '12 at 23:07
Also do you want the numbers set flush left or do you want them aligned on the decimal point? –  David Carlisle May 9 '12 at 23:09
I don't know the difference betewen them but if it is necessary I want in the center of cell –  boutheina May 9 '12 at 23:12
I edit the image I hope that you can understand me now. –  boutheina May 9 '12 at 23:15

format: plain


text:
## A Dime a Dozen

When you were younger did you babysit, deliver papers, or mow lawn(s) for money? If so, you faced stiff competition from other competitors who offered identical services. There was nothing to stop others from offering their services too. All of you charged the “going rate.” If you tried to charge more, your customers would simply buy from someone else. These conditions are very similar to the conditions agricultural growers face.

Figure 8.1. Depending upon the competition and prices offered, a wheat farmer may choose to grow a different crop. (Credit: modification of work by Daniel X. O’Neil/Flickr Creative Commons)

Growing a crop may be more difficult to start than a babysitting or lawn mowing service, but growers face the same fierce competition. In the grand scale of world agriculture, farmers face competition from thousands of others because they sell an identical product. After all, winter wheat is winter wheat. But it is relatively easy for farmers to leave the marketplace for another crop. In this case, they do not sell the family farm, they switch crops.

Take the case of the upper Midwest region of the United States—for many generations the area was called “King Wheat.” According to the United States Department of Agriculture National Agricultural Statistics Service, statistics by state, in 1997, 11.6 million acres of wheat and 780,000 acres of corn were planted in North Dakota. In the intervening 15 or so years has the mix of crops changed? Since it is relatively easy to switch crops, did farmers change what was planted as the relative crop prices changed? We will find out at module’s end.

In the meantime, let’s consider the topic of this module—the perfectly competitive market. This is a market in which entry and exit are relatively easy and competitors are “a dime a dozen.”

## Introduction to Perfect Competition

All businesses face two realities: no one is required to buy their products, and even customers who might want those products may buy from other businesses instead. Firms that operate in perfectly competitive markets face this reality. In this module you will learn how such firms make decisions about how much to produce, how much profit they make, whether to stay in business or not, and many others. Industries differ from one another in terms of how many sellers there are in a specific market, how easy or difficult it is for a new firm to enter, and the type of products that are sold. This is referred to as the market structureHome of the industry. In this module we focus on perfect competition. However, in other modules we will examine other industry types: Monopoly and Monopolistic Competition and Oligopoly.

### Perfect Competition and Why It Matters

Firms are said to be in perfect competitionHome when the following conditions occur: (1) many firms produce identical products; (2) many buyers are available to buy the product, and many sellers are available to sell the product; (3) sellers and buyers have all relevant information to make rational decisions about the product being bought and sold; and (4) firms can enter and leave the market without any restrictions—in other words, there is free entry and exit into and out of the market.

perfectly competitive firmHome is known as a price takerHome, because the pressure of competing firms forces them to accept the prevailing equilibrium price in the market. If a firm in a perfectly competitive market raises the price of its product by so much as a penny, it will lose all of its sales to competitors. When a wheat grower wants to know what the going price of wheat is, he or she has to go to the computer or listen to the radio to check. The market price is determined solely by supply and demand in the entire market and not the individual farmer. Also, a perfectly competitive firm must be a very small player in the overall market, so that it can increase or decrease output without noticeably affecting the overall quantity supplied and price in the market.

A perfectly competitive market is a hypothetical extreme; however, producers in a number of industries do face many competitor firms selling highly similar goods, in which case they must often act as price takers. Agricultural markets are often used as an example. The same crops grown by different farmers are largely interchangeable. According to the United States Department of Agriculture monthly reports, in 2012, U.S. corn farmers received an average price of $6.07 per bushel and wheat farmers received an average price of$7.60 per bushel. A corn farmer who attempted to sell at $7.00 per bushel, or a wheat grower who attempted to sell for$8.00 per bushel, would not have found any buyers. A perfectly competitive firm will not sell below the equilibrium price either. Why should they when they can sell all they want at the higher price? Other examples of agricultural markets that operate in close to perfectly competitive markets are small roadside produce markets and small organic farmers.

Visit this website that reveals the current value of various commodities.

This module examines how profit-seeking firms decide how much to produce in perfectly competitive markets. Such firms will analyze their costs as discussed in the module on Cost and Industry Structure. In the short run, the perfectly competitive firm will seek the quantity of output where profits are highest or, if profits are not possible, where losses are lowest. In this example, the “short run” refers to a situation in which firms are producing with one fixed input and incur fixed costs of production. (In the real world, firms can have many fixed inputs.)

In the long run, perfectly competitive firms will react to profits by increasing production. They will respond to losses by reducing production or exiting the market. Ultimately, a long-run equilibrium will be attained when no new firms want to enter the market and existing firms do not want to leave the market, as economic profits have been driven down to zero.

## Key Concepts and Summary

A perfectly competitive firm is a price taker, which means that it must accept the equilibrium price at which it sells goods. If a perfectly competitive firm attempts to charge even a tiny amount more than the market price, it will be unable to make any sales. In a perfectly competitive market there are thousands of sellers, easy entry, and identical products. A short-run production period is when firms are producing with some fixed inputs. Long-run equilibrium in a perfectly competitive industry occurs after all firms have entered and exited the industry and seller profits are driven to zero.

Perfect competition means that there are many sellers, there is easy entry and exiting of firms, products are identical from one seller to another, and sellers are price takers.

## Self Check: Defining Perfect Competition

Answer the question(s) below to see how well you understand the topics covered in the previous section. This short quiz does not count toward your grade in the class, and you can retake it an unlimited number of times.

You’ll have more success on the Self Check if you’ve completed the Reading in this section.

Use this quiz to check your understanding and decide whether to (1) study the previous section further or (2) move on to the next section.

https://assessments.lumenlearning.com/assessments/651


text:
QUALITY
ASSURANCE

## Product Category

• Product Name
• Product Keyword
• Product Model
• Product Summary
• Product Description
• Multi Field Search

Share to:
• 185kW
• 2.7m3
• 19,123kg
High performance more easily and improve the loading bucket bucket loading coefficient; locking clutch torque converter to achieve large margin traction and speed ramp.

Compared with the H series, fuel consumption reduced by 10%, the economic model optimization, the most fuel-efficient and minimal impact on production efficiency.

High performance more easily and improve the loading bucket bucket loading coefficient; locking clutch torque converter to achieve large margin traction and speed ramp.

Compared with the H series, fuel consumption reduced by 10%, the economic model optimization, the most fuel-efficient and minimal impact on production efficiency.

High performance more easily and improve the loading bucket bucket loading coefficient; locking clutch torque converter to achieve large margin traction and speed ramp.

Compared with the H series, fuel consumption reduced by 10%, the economic model optimization, the most fuel-efficient and minimal impact on production efficiency.

High performance more easily and improve the loading bucket bucket loading coefficient; locking clutch torque converter to achieve large margin traction and speed ramp.

Compared with the H series, fuel consumption reduced by 10%, the economic model optimization, the most fuel-efficient and minimal impact on production efficiency.

High performance more easily and improve the loading bucket bucket loading coefficient; locking clutch torque converter to achieve large margin traction and speed ramp.

Compared with the H series, fuel consumption reduced by 10%, the economic model optimization, the most fuel-efficient and minimal impact on production efficiency.

High performance more easily and improve the loading bucket bucket loading coefficient; locking clutch torque converter to achieve large margin traction and speed ramp.

Compared with the H series, fuel consumption reduced by 10%, the economic model optimization, the most fuel-efficient and minimal impact on production efficiency.

High performance more easily and improve the loading bucket bucket loading coefficient; locking clutch torque converter to achieve large margin traction and speed ramp.

Compared with the H series, fuel consumption reduced by 10%, the economic model optimization, the most fuel-efficient and minimal impact on production efficiency.

High performance more easily and improve the loading bucket bucket loading coefficient; locking clutch torque converter to achieve large margin traction and speed ramp.

Compared with the H series, fuel consumption reduced by 10%, the economic model optimization, the most fuel-efficient and minimal impact on production efficiency.

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# Tangent Function

Go back to  'Trigonometry Basics'

## What is Tangent Function?

The tangent function is a periodic function which is very important in trigonometry.

The simplest way to understand the tangent function is to use the unit circle. In trigonometry, a unit circle is the circle of radius one centered at the origin $$\left( {0,0} \right)$$ in the Cartesian coordinate system.

Now that we have seen how to define trigonometric ratios for arbitrary angles, we are ready to study trigonometric ratios from the perspective of functions. Let's try to understand the Graph of Tangent Function.

## Graph of Tangent Function:

We define the $$\tan$$ function as

$f\left( x \right) = \tan x$

Let us study the variation in the $$\tan$$ function as $$x$$ goes from 0 to $$\frac{\pi }{2}$$. Consider the following figure, which shows $$P$$ moving through the first quadrant:

We make the following observations:

• Initially, when $$x = 0$$, the perpendicular $$PQ = 0$$ while the base $$OQ = 1$$. Thus, $$\tan 0 = 0$$.

• As $$x$$ increases, $$PQ$$ increases, while $$OQ$$ decreases. Thus, $$\tan x = \frac{{PQ}}{{OQ}}$$ increases.

• When $$x = \frac{\pi }{4}$$, $$PQ = OQ$$, so that $$\tan \frac{\pi }{4} = 1$$.

• As $$x$$ increases further, $$PQ$$ increases towards 1, while $$OQ$$ decreases towards 0.

• As $$x$$ nears $$\frac{\pi }{2}$$, $$PQ$$ comes closer and closer to 1, while $$OQ$$ comes closer and closer to 0. Thus, $$\frac{{PQ}}{{OQ}}$$ will become larger and larger. In fact, it will go towards infinity as $$x$$ comes very close to $$\frac{\pi }{2}$$.

• Exactly at $$x = \frac{\pi }{2}$$, $$PQ$$ becomes exactly 1, while $$OQ$$ becomes exactly 0. The ratio $$\frac{{PQ}}{{OQ}}$$ becomes mathematically undefined. Thus, $$\tan \frac{\pi }{2}$$ is undefined.

Making use of these observations, the graph of $$f\left( x \right) = \tan x$$ for $$x$$ in the range $$\left[ {0,\frac{\pi }{2}} \right)$$ is shown below:

Now how the curve shoots up towards infinity as $$x$$ nears $$\frac{\pi }{2}$$. Exactly at $$x = \frac{\pi }{2}$$, the tan function is undefined. The vertical dashed line in the graph highlights the fact that the curve tends towards this line, but never actually manages to touch this line, no matter how close we come to $$x = \frac{\pi }{2}$$.

Let us now study the variation in the $$\tan x$$ function as $$x$$ goes from $$\frac{\pi }{2}$$ to $$\pi$$. The following figure shows $$P$$ moving through the second quadrant:

Note the following observations:

• When $$P$$ just crosses from the $$y$$-axis over into the second quadrant (when $$x$$ increases just slightly above $$\frac{\pi }{2}$$), the perpendicular $$PQ$$ is still very close to 1, while the base $$OQ$$ is still very close to 0. However, $$OQ$$ is now negative, which means that $$\frac{{PQ}}{{OQ}}$$ will be very large in magnitude but negative in sign. Thus, if $$x$$ is larger than $$\frac{\pi }{2}$$ but comes closer and closer to $$\frac{\pi }{2}$$, $$\tan x$$ will shoot off towards negative infinity.

• As $$P$$ moves further into the second quadrant, $$PQ$$ decreases in magnitude, while $$OQ$$ increases in magnitude (but is still negative). Thus, $$\frac{{PQ}}{{OQ}}$$ decreases in magnitude but stays negative.

• When $$x = \frac{{3\pi }}{4}$$ (1350), $$\left| {PQ} \right| = \left| {OQ} \right|$$, so $$\tan \frac{{3\pi }}{4} = - 1$$.

• As $$P$$ moves towards the position of $$180^\circ$$ or $$\pi$$ radians, $$PQ$$ comes closer to 0 while $$\left| {OQ} \right|$$ comes closer to 1. Thus, the ratio $$\frac{{PQ}}{{OQ}}$$ comes closer to 0 (but stays negative all this while).

• When $$x$$ is exactly $$\pi$$, $$PQ = 0$$ while $$OQ = - 1$$, so $$\tan \pi = 0$$.

We make use of these observations to extend the graph of $$f\left( x \right) = \tan x$$ to include $$x$$ in the interval $$\left( {\frac{\pi }{2},\pi } \right]$$ :

Now how the curve shoots down towards negative infinity as $$x$$ nears $$\frac{\pi }{2}$$ from the right side.

To understand this behavior around $$\frac{\pi }{2}$$ better, consider the following figure, which shows two positions of $$P$$, one when $$x$$ is just less than $$\frac{\pi }{2}$$, and the other when $$x$$ is just greater than $$\frac{\pi }{2}$$:

For $${P_1},\,{P_1}{Q_1}$$ will be very near to 1, and $$O{Q_1}$$ will be positive and very small in magnitude (near to 0). Thus, $$\frac{{{P_1}{Q_1}}}{{O{Q_1}}}$$ will be very large (it will approach positive infinity as $$x$$ comes even closer to $$\frac{\pi }{2}$$ from the lower side).

Now, consider the second position $${P_2}$$. $${P_2}{Q_2}$$ will also be very near to 1, and $$O{Q_2}$$ will again be close to 0 in magnitude, but $$O{Q_2}$$ will be negative. Thus, $$\frac{{{P_2}{Q_2}}}{{O{Q_2}}}$$ will be very large in magnitude but negative in sign (it will approach negative infinity as $$x$$ comes even closer to $$\frac{\pi }{2}$$ from the upper side).

Thus, as $$x$$ goes from slightly below $$\frac{\pi }{2}$$ to slightly above $$\frac{\pi }{2}$$, the curve of $$f\left( x \right) = \tan x$$ jumps from nearing positive infinity to nearing negative infinity, through a discontinuity at $$x = \frac{\pi }{2}$$.

It is not difficult to see that as $$P$$ moves further, the variation in $$\tan x$$ will start to repeat. As $$x$$ increases from $$\pi$$ to $$\frac{{3\pi }}{2}$$, the variation in $$\tan x$$ will be the same as the variation in $$\tan x$$ for the range $$\left[ {0,\frac{\pi }{2}} \right)$$.

As $$x$$ increases from $$\frac{{3\pi }}{2}$$ to $$2\pi$$, the variation in $$\tan x$$ will be the same as the variation in $$\tan x$$ for the range $$\left( {\frac{\pi }{2},\pi } \right]$$. Make sure you understand why this is so.

Thus, we can now extend our graph to obtain the complete graph of $$f\left( x \right) = \tan x$$:

We make the following observations:

• The $$\tan x$$ functions is not defined for all odd multiples of $$\frac{\pi }{2}$$. This is because the base is 0 for these positions. Thus, the domain of the $$\tan x$$ function is $$\mathbb{R}\ - \left\{ {\left( {2n + 1} \right)\frac{\pi }{2}} \right\}$$, where $$n$$ belongs to the set of integers.

• The range of this function is $$\mathbb{R}$$, as the $$y$$-variation in the graph is from negative infinity to positive infinity.

• The $$\tan x$$ function has a cycle of $$\pi$$, instead of $$2\pi$$.

Note: Domain of the $$\tan x$$ function is $$\mathbb{R}\ - \left\{ {\left( {2n + 1} \right)\frac{\pi }{2}} \right\}$$, where $$n$$ belongs to the set of integers and the range of $$\tan x$$ function is $$\mathbb{R}$$.

## Solved Example:

Example 1: Find all values of $$x$$ such that $$\tan x = 1$$.

Solution: The following figure shows the intersection of $$y = \tan x$$ and $$y = 1$$:

In the interval $$\left[ {0,\pi } \right]$$ , we find one solution: $$x = \frac{\pi }{4}$$. Since $$\tan x$$ has a cycle of $$\pi$$, we can write the set of all solutions as:

$\boxed{x = n\pi + \frac{\pi }{4},\,\,\,n \in \mathbb{Z}}$

Challenge: Find the domain of the following function:

$f(x) = \frac{1}{{1 - \tan x}}$

Tip: $$f(x)$$ will be undefined when $${1 - \tan x = 0}$$.


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# Kneser's theorem (combinatorics)

Kneser's theorem (combinatorics) In the branch of mathematics known as additive combinatorics, Kneser's theorem can refer to one of several related theorems regarding the sizes of certain sumsets in abelian groups. These are named after Martin Kneser, who published them in 1953[1] and 1956.[2] They may be regarded as extensions of the Cauchy–Davenport theorem, which also concerns sumsets in groups but is restricted to groups whose order is a prime number.[3] The first three statements deal with sumsets whose size (in various senses) is strictly smaller than the sum of the size of the summands. The last statement deals with the case of equality for Haar measure in connected compact abelian groups.

Contents 1 Strict inequality 1.1 Cardinality 1.2 Lower asymptotic density in the natural numbers 1.3 Haar measure in locally compact abelian (LCA) groups 2 Equality in connected compact abelian groups 3 Notes 4 References Strict inequality If {displaystyle G} is an abelian group and {displaystyle C} is a subset of {displaystyle G} , the group {displaystyle H(C):={gin G:g+C=C}} is the stabilizer of {displaystyle C} .

Cardinality Let {displaystyle G} be an abelian group. If {displaystyle A} and {displaystyle B} are nonempty finite subsets of {displaystyle G} satisfying {displaystyle |A+B|<|A|+|B|} and {displaystyle H} is the stabilizer of {displaystyle A+B} , then {displaystyle {begin{aligned}|A+B|&=|A+H|+|B+H|-|H|.end{aligned}}} This statement is a corollary of the statement for LCA groups below, obtained by specializing to the case where the ambient group is discrete. A self-contained proof is provided in Nathanson's textbook.[4] Lower asymptotic density in the natural numbers The main result of Kneser's 1953 article[1] is a variant of Mann's theorem on Schnirelmann density. If {displaystyle C} is a subset of {displaystyle mathbb {N} } , the lower asymptotic density of {displaystyle C} is the number {displaystyle {underline {d}}(C):=liminf _{nto infty }{frac {|Ccap {1,dots ,n}|}{n}}} . Kneser's theorem for lower asymptotic density states that if {displaystyle A} and {displaystyle B} are subsets of {displaystyle mathbb {N} } satisfying {displaystyle {underline {d}}(A+B)<{underline {d}}(A)+{underline {d}}(B)} , then there is a natural number {displaystyle k} such that {displaystyle H:=kmathbb {N} cup {0}} satisfies the following two conditions: {displaystyle (A+B+H)setminus (A+B)} is finite, and {displaystyle {underline {d}}(A+B)={underline {d}}(A+H)+{underline {d}}(B+H)-{underline {d}}(H).} Note that {displaystyle A+Bsubseteq A+B+H} , since {displaystyle 0in H} . Haar measure in locally compact abelian (LCA) groups Let {displaystyle G} be an LCA group with Haar measure {displaystyle m} and let {displaystyle m_{*}} denote the inner measure induced by {displaystyle m} (we also assume {displaystyle G} is Hausdorff, as usual). We are forced to consider inner Haar measure, as the sumset of two {displaystyle m} -measurable sets can fail to be {displaystyle m} -measurable. Satz 1 of Kneser's 1956 article[2] can be stated as follows: If {displaystyle A} and {displaystyle B} are nonempty {displaystyle m} -measurable subsets of {displaystyle G} satisfying {displaystyle m_{*}(A+B)0,m(B)>0} , and equation (1), then there is a continuous surjective homomorphism {displaystyle phi :Gto mathbb {T} } and there are closed intervals {displaystyle I} , {displaystyle J} in {displaystyle mathbb {T} } such that {displaystyle Asubseteq phi ^{-1}(I)} , {displaystyle Bsubseteq phi ^{-1}(J)} , {displaystyle m(A)=m(phi ^{-1}(I))} , and {displaystyle m(B)=m(phi ^{-1}(J))} .

Notes ^ Jump up to: a b Kneser, Martin (1953). "Abschätzungen der asymptotischen Dichte von Summenmengen". Math. Z. (in German). 58: 459–484. doi:10.1007/BF01174162. S2CID 120456416. Zbl 0051.28104. ^ Jump up to: a b c Kneser, Martin (1956). "Summenmengen in lokalkompakten abelschen Gruppen". Math. Z. (in German). 66: 88–110. doi:10.1007/BF01186598. S2CID 120125011. Zbl 0073.01702. ^ Geroldinger & Ruzsa (2009, p. 143) ^ Nathanson, Melvyn B. (1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics. Vol. 165. Springer-Verlag. pp. 109–132. ISBN 0-387-94655-1. Zbl 0859.11003. References Geroldinger, Alfred; Ruzsa, Imre Z., eds. (2009). Combinatorial number theory and additive group theory. Advanced Courses in Mathematics CRM Barcelona. Elsholtz, C.; Freiman, G.; Hamidoune, Y. O.; Hegyvári, N.; Károlyi, G.; Nathanson, M.; Solymosi, J.; Stanchescu, Y. With a foreword by Javier Cilleruelo, Marc Noy and Oriol Serra (Coordinators of the DocCourse). Basel: Birkhäuser. ISBN 978-3-7643-8961-1. Zbl 1177.11005. Grynkiewicz, David (2013). Structural Additive Theory. Developments in Mathematics. Vol. 30. Springer. p. 61. ISBN 978-3-319-00415-0. Zbl 1368.11109. Tao, Terence; Vu, Van H. (2010), Additive Combinatorics, Cambridge: Cambridge University Press, ISBN 978-0-521-13656-3, Zbl 1179.11002 Categories: Theorems in combinatoricsSumsets

Si quieres conocer otros artículos parecidos a Kneser's theorem (combinatorics) puedes visitar la categoría Sumsets.

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# Example of a hardness-of-approximation proof which improves the approximation factor?

Are there any examples of hardness-of-approximation reductions where we get a better hardness bound for the problem we've reduced to than the problem we've reduced from? In the examples I've seen so far, the bound is usually degraded in the reduction.

• A lot of reductions amplify hardness by toying with the objective function. For example, Min Circuit SAT (given a Boolean circuit, what is the smallest hamming-weight input that evaluates to true) is NP-hard because it can encode min vertex cover. This gives a small hardness factor of ~2. So it's hard to distinguish instances that can be solved with weight a=f(n) from those with weight 1.5a. But now we can reduce from this circuit sat instance by making the circuit that accepts either solutions to the original CSAT instance of weight <= a, or anything of weight > n/2, for $\omega(1)$ hardness. – Yonatan N Jun 20 '19 at 19:01
• @YonatanN That's a great example. I would accept it as an answer if you want to copy paste it. – Elliot Gorokhovsky Jun 23 '19 at 14:16


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# Plotting summary statistics with mean, sd, min and max?

I am from an economics background and usually in the discipline the summary statistics of the variables are reported in a table. However, I wish to plot them.

I could modify a box plot to allow it to display the mean, standard deviation, minimum and maximum but I don't wish to do so as box plots are traditionally used to display medians and Q1 and Q3.

All of my variables have different scales. It would be great if somebody could suggest a meaningful way by which I could plot these summary statistics. I can work with R or Stata.

• Welcome to the list. If you are asking about R commands then this question is off-topic here. But it seems you are asking primarily about what a good plot would look like and secondarily about how to create it. If so, I suggest deleting "with R" from your title and maybe stating, in the body, that you have R available. – Peter Flom Nov 27 '13 at 11:38

There is a reason why Tukey's boxplot is universal, it can be applied to data derived from different distributions, from Gaussian to Poisson, etc. Median, MAD (median absolute deviation) or IQR (interquartile range) are more robust measures when data deviates from normality. However, mean and SD are more prone to outliers, and they should be interpreted with respect to the underlying distribution. The solution below is more suitable for normal or log-normal data. You may browse through a selection of robust measures here, and explore the WRS R package here.

# simulating dataset
set.seed(12)
d1 <- rnorm(100, sd=30)
d2 <- rnorm(100, sd=10)
d <- data.frame(value=c(d1,d2), condition=rep(c("A","B"),each=100))

# function to produce summary statistics (mean and +/- sd), as required for ggplot2
data_summary <- function(x) {
mu <- mean(x)
sigma1 <- mu-sd(x)
sigma2 <- mu+sd(x)
return(c(y=mu,ymin=sigma1,ymax=sigma2))
}

# require(ggplot2)
ggplot(data=d, aes(x=condition, y=value, fill=condition)) +
geom_crossbar(stat="summary", fun.y=data_summary, fun.ymax=max, fun.ymin=min)

Additionally by adding + geom_jitter() or + geom_point() to the code above you can simultaneously visualise the raw data values.

Thanks to @Roland for pointing out the violin plot. It has an advantage in visualising probability density at the same time as summary statistic:

# require(ggplot2)
ggplot(data=d, aes(x=condition, y=value, fill=condition)) +
geom_violin() + stat_summary(fun.data=data_summary)

Both examples are shown below.

• I'd prefer a violin plot over this. – Roland Nov 27 '13 at 15:42
• Depending on the purpose of the analysis, mean and standard deviation is exactly what you need. I don't understand however the inconsistancy in R's summary.data.frame. It shows means but no sds. I can't think of many situations where means are useful but standard deviations misleading. – Michael M Nov 27 '13 at 16:31
• Indeed, sometimes you need to see the mean and the SD so that you judge whether they are useful.... – Nick Cox Nov 27 '13 at 17:37
• @TWL: The topic is much too broad to discuss here. But take for instance economic evaluations of drugs: For the patient, maybe it's important to know median treatment duration, while for the health insurance company it's the mean treatment duration because they need to pay it for each patient. A curious fact: In the case of the exponential distribution, mean +/- 1 standard deviation covers 68% of all mass, mean +/- 2 sds covers about 95% of all mass. As for the normal. (But it's mere chance ;)) – Michael M Nov 27 '13 at 18:04
• Thanks all, I like the proposed violin plots so will go ahead with that choice :-) – Ridhima Nov 28 '13 at 15:11

One option I have seen used which avoids confusion with boxplots (assuming you have medians or original data available) is to plot a boxplot and add a symbol that marks the mean (hopefully with a legend to make this explicit). This version of the boxplot that adds a marker for the mean is mentioned, for example in Frigge et al (1989) [1] :

The left plot shows a + symbol as a mean marker and the right plot uses a triangle at the edge, adapting the mean marker from Doane & Tracy's beam-and-fulcrum plot [2].

If you don't have (or really don't want to show) the median a new plot will be needed and then it would be good for it to be visually distinct from a boxplot.

Perhaps something like this:

... which plots the minimum, maximum, mean and mean $\pm$ sd for each sample using different symbols and then draws a rectangle, or perhaps better, something like this:

... which plots the minimum, maximum, mean and mean $\pm$ sd for each sample using different symbols and then draws a line (in fact at present that's actually a rectangle as before, but drawn narrow; it should be changed to drawing a line)

If your numbers are on very different scales, but are all positive, you might consider working with logs, or you might do small multiples with different (but clearly marked) scales

Code ( presently not particularly 'nice' code, but at the moment this is just exploring ideas, it's not a tutorial on writing good R code):

fivenum.ms=function(x) {r=range(x);m=mean(x);s=sd(x);c(r[1],m-s,m,m+s,r[2])}
eps=.015

plot(factor(c(1,2)),range(c(A,B)),type="n",border=0)
points((rep(c(1,2),each=5)),c(fivenum.ms(A),fivenum.ms(B)),col=rep(c(2,4),each=5),pch=rep(c(1,16,9,16,1),2),ylim=c(range(A,B)),cex=1.2,lwd=2,xlim=c(0.5,2.5),ylab="",xlab="")
rect(1-1.2*eps,fivenum.ms(A)[2],1+1.4*eps,fivenum.ms(A)[4],lwd=2,col=2,den=0)
rect(2-1.2*eps,fivenum.ms(B)[2],2+1.4*eps,fivenum.ms(B)[4],lwd=2,col=4,den=0)

plot(factor(c(1,2)),range(c(A,B)),type="n",border=0)
points((rep(c(1,2),each=5)),c(fivenum.ms(A),fivenum.ms(B)),col=rep(c(2,4),each=5),pch=rep(c(1,16,9,16,1),2),ylim=c(range(A,B)),cex=1.2,lwd=2,xlim=c(0.5,2.5),ylab="",xlab="")
rect(1-eps/9,fivenum.ms(A)[2],1+eps/3,fivenum.ms(A)[4],lwd=2,col=2,den=0)
rect(2-eps/9,fivenum.ms(B)[2],2+eps/3,fivenum.ms(B)[4],lwd=2,col=4,den=0)

[1] Frigge, M., D. C. Hoaglin, and B. Iglewicz (1989),
"Some implementations of the box plot."
American Statistician, 43 (Feb): 50-54.

[2] Doane D.P. and R.L. Tracy (2000),
"Using Beam and Fulcrum Displays to Explore Data"
American Statistician, 54(4):289–290, November


text:
# Search by Topic

#### Resources tagged with Mathematical reasoning & proof similar to Roots and Coefficients:

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### There are 184 results

Broad Topics > Using, Applying and Reasoning about Mathematics > Mathematical reasoning & proof

### Proof Sorter - Quadratic Equation

##### Stage: 4 and 5 Challenge Level:

This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.

### Golden Eggs

##### Stage: 5 Challenge Level:

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

### Pent

##### Stage: 4 and 5 Challenge Level:

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

### Plus or Minus

##### Stage: 5 Challenge Level:

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

### Direct Logic

##### Stage: 5 Challenge Level:

Can you work through these direct proofs, using our interactive proof sorters?

##### Stage: 5 Challenge Level:

Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.

### Target Six

##### Stage: 5 Challenge Level:

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

### Proof of Pick's Theorem

##### Stage: 5 Challenge Level:

Follow the hints and prove Pick's Theorem.

### Whole Number Dynamics I

##### Stage: 4 and 5

The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.

### Where Do We Get Our Feet Wet?

##### Stage: 5

Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.

### Whole Number Dynamics III

##### Stage: 4 and 5

In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.

### Whole Number Dynamics II

##### Stage: 4 and 5

This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.

### Pair Squares

##### Stage: 5 Challenge Level:

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

### Long Short

##### Stage: 4 Challenge Level:

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

### Janine's Conjecture

##### Stage: 4 Challenge Level:

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

### Leonardo's Problem

##### Stage: 4 and 5 Challenge Level:

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

### Proof Sorter - Geometric Series

##### Stage: 5 Challenge Level:

This is an interactivity in which you have to sort into the correct order the steps in the proof of the formula for the sum of a geometric series.

### Napoleon's Hat

##### Stage: 5 Challenge Level:

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

##### Stage: 4 Challenge Level:

Four jewellers share their stock. Can you work out the relative values of their gems?

### Square Mean

##### Stage: 4 Challenge Level:

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

### Binomial

##### Stage: 5 Challenge Level:

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

### Yih or Luk Tsut K'i or Three Men's Morris

##### Stage: 3, 4 and 5 Challenge Level:

Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .

### Continued Fractions II

##### Stage: 5

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

### The Great Weights Puzzle

##### Stage: 4 Challenge Level:

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?

### Multiplication Square

##### Stage: 4 Challenge Level:

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

### The Golden Ratio, Fibonacci Numbers and Continued Fractions.

##### Stage: 4

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.

### Iffy Logic

##### Stage: 4 and 5 Challenge Level:

Can you rearrange the cards to make a series of correct mathematical statements?

### Contrary Logic

##### Stage: 5 Challenge Level:

Can you invert the logic to prove these statements?

### Dodgy Proofs

##### Stage: 5 Challenge Level:

These proofs are wrong. Can you see why?

### The Clue Is in the Question

##### Stage: 5 Challenge Level:

This problem is a sequence of linked mini-challenges leading up to the proof of a difficult final challenge, encouraging you to think mathematically. Starting with one of the mini-challenges, how. . . .

### Notty Logic

##### Stage: 5 Challenge Level:

Have a go at being mathematically negative, by negating these statements.

### Thousand Words

##### Stage: 5 Challenge Level:

Here the diagram says it all. Can you find the diagram?

### Water Pistols

##### Stage: 5 Challenge Level:

With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?

### Euclid's Algorithm II

##### Stage: 5

We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.

### To Prove or Not to Prove

##### Stage: 4 and 5

A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.

### The Frieze Tree

##### Stage: 3 and 4

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

### Impossible Sandwiches

##### Stage: 3, 4 and 5

In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.

### The Triangle Game

##### Stage: 3 and 4 Challenge Level:

Can you discover whether this is a fair game?

### Unit Interval

##### Stage: 4 and 5 Challenge Level:

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

### Matter of Scale

##### Stage: 4 Challenge Level:

Prove Pythagoras' Theorem using enlargements and scale factors.

### Modulus Arithmetic and a Solution to Differences

##### Stage: 5

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.

##### Stage: 5 Challenge Level:

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

### Proof Sorter - Sum of an AP

##### Stage: 5 Challenge Level:

Use this interactivity to sort out the steps of the proof of the formula for the sum of an arithmetic series. The 'thermometer' will tell you how you are doing

### An Introduction to Number Theory

##### Stage: 5

An introduction to some beautiful results of Number Theory

### Tree Graphs

##### Stage: 5 Challenge Level:

A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree has. . . .

### Diverging

##### Stage: 5 Challenge Level:

Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.

### Middle Man

##### Stage: 5 Challenge Level:

Mark a point P inside a closed curve. Is it always possible to find two points that lie on the curve, such that P is the mid point of the line joining these two points?

### Pythagorean Golden Means

##### Stage: 5 Challenge Level:

Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.

### Sperner's Lemma

##### Stage: 5

An article about the strategy for playing The Triangle Game which appears on the NRICH site. It contains a simple lemma about labelling a grid of equilateral triangles within a triangular frame.


text:
• Browse all
Evidence for Collective Multiparticle Correlations in p-Pb Collisions

The collaboration
Phys.Rev.Lett. 115 (2015) 012301, 2015

Abstract (data abstract)
CERN-LHC. The second-order azimuthal anisotropy Fourier harmonics, $v_2$, are obtained in pPb and PbPb collisions over a wide pseudorapidity ($\eta$) range based on correlations among six or more charged particles. The pPb data, corresponding to an integrated luminosity of 35 inverse nanobarns, were collected during the 2013 LHC pPb run at a nucleon-nucleon center-of-mass energy of 5.02 TeV by the CMS experiment. A sample of semi-peripheral PbPb collision data at \sqrt{s_{NN}}= 2.76 TeV, corresponding to an integrated luminosity of 2.5 inverse microbarns and covering a similar range of particle multiplicities as the pPb data, is also analyzed for comparison. The six- and eight-particle cumulant and the Lee-Yang zeros methods are used to extract the $v_2$ coefficients, extending previous studies of two- and four-particle correlations. For both the pPb and PbPb systems, the $v_2$ values obtained with correlations among more than four particles are consistent with previously published four-particle results. These data support the interpretation of a collective origin for the previously observed long-range (large $\Delta\eta$) correlations in both systems. The ratios of $v_2$ values corresponding to correlations including different numbers of particles are compared to theoretical predictions that assume a hydrodynamic behavior of a pPb system dominated by fluctuations in the positions of participant nucleons. These results provide new insights into the multi-particle dynamics of collision systems with a very small overlapping region.

• #### Table 1

Data from Figure 1

10.17182/hepdata.67530.v1/t1

The cumulant $c_2\{6\}$ extracted for all charged particles with $0.3 < p_T < 3.0$ GeV/c as a function of $N_{trk}^{offline}$...

• #### Table 2

Data from Figure 1

10.17182/hepdata.67530.v1/t2

The cumulant $c_2\{8\}$ extracted for all charged particles with $0.3 < p_T < 3.0$ GeV/c as a function of $N_{trk}^{offline}$...

• #### Table 3

Data from Figure 1

10.17182/hepdata.67530.v1/t3

The cumulant $c_2\{6\}$ extracted for all charged particles with $0.3 < p_T < 3.0$ GeV/c as a function of $N_{trk}^{offline}$...

• #### Table 4

Data from Figure 1

10.17182/hepdata.67530.v1/t4

The cumulant $c_2\{8\}$ extracted for all charged particles with $0.3 < p_T < 3.0$ GeV/c as a function of $N_{trk}^{offline}$...

• #### Table 5

Data from Figure 2

10.17182/hepdata.67530.v1/t5

The elliptic flow $v_2\{6\}$ extracted for all charged particles with $0.3 < p_T < 3.0$ GeV/c as a function of...

• #### Table 6

Data from Figure 2

10.17182/hepdata.67530.v1/t6

The elliptic flow $v_2\{8\}$ extracted for all charged particles with $0.3 < p_T < 3.0$ GeV/c as a function of...

• #### Table 7

Data from Figure 2

10.17182/hepdata.67530.v1/t7

The elliptic flow $v_2\{\text{LYZ}\}$ extracted for all charged particles with $0.3 < p_T < 3.0$ GeV/c as a function of...

• #### Table 8

Data from Figure 2

10.17182/hepdata.67530.v1/t8

The elliptic flow $v_2\{6\}$ extracted for all charged particles with $0.3 < p_T < 3.0$ GeV/c as a function of...

• #### Table 9

Data from Figure 2

10.17182/hepdata.67530.v1/t9

The elliptic flow $v_2\{8\}$ extracted for all charged particles with $0.3 < p_T < 3.0$ GeV/c as a function of...

• #### Table 10

Data from Figure 2

10.17182/hepdata.67530.v1/t10

The elliptic flow $v_2\{\text{LYZ}\}$ extracted for all charged particles with $0.3 < p_T < 3.0$ GeV/c as a function of...

• #### Table 11

Data from Figure 3

10.17182/hepdata.67530.v1/t11

The cumulant ratio $c_2\{6\}/v_2\{4\}$ as a function of $c_2\{4\}/v_2\{2\}$ in pPb collisions.

• #### Table 12

Data from Figure 3

10.17182/hepdata.67530.v1/t12

The cumulant ratio $c_2\{8\}/v_2\{6\}$ as a function of $c_2\{4\}/v_2\{2\}$ in pPb collisions.

• #### Table 13

Data from Figure 3

10.17182/hepdata.67530.v1/t13

The cumulant ratio $c_2\{6\}/v_2\{4\}$ as a function of $c_2\{4\}/v_2\{2\}$ in PbPb collisions.

• #### Table 14

Data from Figure 3

10.17182/hepdata.67530.v1/t14

The cumulant ratio $c_2\{8\}/v_2\{6\}$ as a function of $c_2\{4\}/v_2\{2\}$ in PbPb collisions.


text:
# INSPIRE search

VALUE (MeV) EVTS DOCUMENT ID TECN CHG  COMMENT
$\bf{ 50\text{ to }100 }$ OUR ESTIMATE  Width determination very model dependent. Peak width in ${{\mathit \eta}}{{\mathit \pi}}$ is about 60 MeV, but decay width can be much larger.
• • • We do not use the following data for averages, fits, limits, etc. • • •
$75.6$ $\pm1.6$ ${}^{+17.4}_{-10.0}$ 1
 2009 A
BELL ${{\mathit \gamma}}$ ${{\mathit \gamma}}$ $\rightarrow$ ${{\mathit \pi}^{0}}{{\mathit \eta}}$
$80.2$ $\pm3.8$ $\pm5.4$ 2
 2008 A
RVUE 0 ${{\overline{\mathit p}}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit \pi}^{0}}{{\mathit \pi}^{0}}{{\mathit \eta}}$
$50$ $\pm13$ $\pm4$ 318
 2002 B
L3 $183 - 209$ ${{\mathit e}^{+}}$ ${{\mathit e}^{-}}$ ${{\mathit e}^{-}}$ ${{\mathit \eta}}$ ${{\mathit \pi}^{+}}$ ${{\mathit \pi}^{-}}$
$72$ $\pm16$
 2000 H
450 ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit p}_{{f}}}{{\mathit \eta}}{{\mathit \pi}^{0}}{{\mathit p}_{{s}}}$
$61$ $\pm19$
 2000 H
450 ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit \Delta}_{{f}}^{++}}{{\mathit \eta}}{{\mathit \pi}^{-}}{{\mathit p}_{{s}}}$
$\sim{}\text{ 42}$ 3
 1999
RVUE ${{\mathit \eta}}{{\mathit \pi}}$ , ${{\mathit K}}{{\overline{\mathit K}}}$
$\sim{}\text{ 112}$ 3
 1999 B
RVUE ${{\mathit \pi}}$ ${{\mathit \pi}}$ $\rightarrow$ ${{\mathit \eta}}{{\mathit \pi}}$ , ${{\mathit K}}{{\overline{\mathit K}}}$
$71$ $\pm7$
 1999
B852 $18.3$ ${{\mathit \pi}^{-}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit \eta}}{{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit n}}$
$92$ $\pm20$ 3
 1998 B
RVUE Compilation
$65$ $\pm10$ 4
 1998 B
OBLX $\pm{}$ 0.0 ${{\overline{\mathit p}}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit K}^{\pm}}{{\mathit K}_{{s}}}{{\mathit \pi}^{\mp}}$
$\sim{}\text{ 100}$
 1996
RVUE ${{\mathit \pi}}$ ${{\mathit \pi}}$ $\rightarrow$ ${{\mathit \pi}}{{\mathit \pi}}$ , ${{\mathit K}}{{\overline{\mathit K}}}$ , ${{\mathit K}}{{\mathit \pi}}$ , ${{\mathit \eta}}{{\mathit \pi}}$
$202$
 1995
RVUE ${{\mathit \eta}}$ ${{\mathit \pi}}$ $\rightarrow$ ${{\mathit \eta}}{{\mathit \pi}}$ , ${{\mathit K}}{{\overline{\mathit K}}}$ , ${{\mathit K}}{{\mathit \pi}}$ , ${{\mathit \eta}}{{\mathit \pi}}$
$54.12$ $\pm0.34$ $\pm0.12$
 1994 C
CBAR $0.0$ ${{\overline{\mathit p}}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit \omega}}{{\mathit \eta}}{{\mathit \pi}^{0}}$
$54$ $\pm10$ 5
 1992
CBAR 0.0 ${{\overline{\mathit p}}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit \eta}}{{\mathit \eta}}{{\mathit \pi}^{0}}$
$95$ $\pm14$ 1040 5
 1991 B
OMEG $\pm{}$ 300 ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit p}}{{\mathit p}}{{\mathit \eta}}{{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$
$62$ $\pm15$ 500 6
 1981
OMEG $\pm{}$ 12 ${{\mathit \pi}^{-}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit \eta}}{{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit \pi}^{-}}{{\mathit p}}$
$60$ $\pm20$ 145 6
 1979
HBC $\pm{}$ 4.2 ${{\mathit K}^{-}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit \Lambda}}{{\mathit \eta}}$2 ${{\mathit \pi}}$
$60$ ${}^{+50}_{-30}$ 47
 1978
OSPK - 4.5 ${{\mathit \pi}^{-}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit p}}{{\mathit X}^{-}}$
$86.0$ ${}^{+60.0}_{-50.0}$ 50
 1978
OMEG $\pm{}$ 12$-$15 ${{\mathit \pi}^{-}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit n}}{{\mathit \eta}}$2 ${{\mathit \pi}}$
$44$ $\pm22$
 1977
HBC - 16 ${{\mathit \pi}^{\mp}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit p}}{{\mathit \eta}}$3 ${{\mathit \pi}}$
$80\text{ to }300$ 7
 1976
RVUE - 4.2 ${{\mathit K}^{-}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit \Lambda}}{{\mathit \eta}}$2 ${{\mathit \pi}}$
$16.0$ ${}^{+25.0}_{-16.0}$ 70
 1975
HBC - 3.1$-$6 ${{\mathit K}^{-}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit \Lambda}}{{\mathit \eta}}$2 ${{\mathit \pi}}$
$30$ $\pm5$ 150
 1972
HBC $\pm{}$ 0.7 ${{\overline{\mathit p}}}$ ${{\mathit p}}$ $\rightarrow$ 7 ${{\mathit \pi}}$
$40$ $\pm15$
 1969
DBC $\pm{}$ 2.7 ${{\mathit \pi}^{+}}{{\mathit d}}$
$60$ $\pm30$ 15
 1969 B
HBC - 4.5 ${{\mathit K}^{-}}$ ${{\mathit N}}$ $\rightarrow$ ${{\mathit \eta}}{{\mathit \pi}}{{\mathit \Lambda}}$
$80$ $\pm30$ 30
 1968
HBC $\pm{}$ 5.5 ${{\mathit K}^{-}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit \Lambda}}{{\mathit \eta}}$2 ${{\mathit \pi}}$
1  From a fit with the S-wave amplitude including two interfering Breit-Wigners plus a background term.
2  From the T-matrix pole on sheet II, using AMSLER 1994D and ABELE 1998 .
3  T-matrix pole.
4  The ${{\mathit \eta}}{{\mathit \pi}}$ width.
5  From a single Breit-Wigner fit.
6  From ${{\mathit f}_{{1}}{(1285)}}$ decay.
7  Using a two-channel resonance parametrization of GAY 1976B data.
References:
 UEHARA 2009A
PR D80 032001 High-Statistics Study of ${{\mathit \eta}}{{\mathit \pi}^{0}}$ Production in Two-Photon Collisions
 BUGG 2008A
PR D78 074023 Reanalysis of Data on ${{\mathit a}_{{0}}{(1450)}}$ and ${{\mathit a}_{{0}}{(980)}}$
 ACHARD 2002B
PL B526 269 ${{\mathit f}_{{1}}{(1285)}}$ Formation in Two Photon Collisions at LEP
 BARBERIS 2000H
PL B488 225 A Study of the Centrally Produced ${{\mathit \eta}}{{\mathit \pi}^{0}}$ and ${{\mathit \eta}}{{\mathit \pi}^{-}}$ Systems in ${{\mathit p}}{{\mathit p}}$ Interactions at 450 ${\mathrm {GeV/}}\mathit c$
 OLLER 1999
PR D60 099906 (erratum) Meson-Meson Interaction in a Nonperturbative Chiral Approach
 OLLER 1999B
NP A652 407 (erratum) Chiral Symmetry Amplitudes in the S-wave Isoscalar and Isovector Channels and the ${{\mathit \sigma}}$, ${{\mathit f}_{{0}}{(980)}}$, ${{\mathit a}_{{0}}{(980)}}$ Scalar Mesons
 TEIGE 1999
PR D59 012001 Properties of the ${{\mathit a}_{{0}}{(980)}}$ Meson
 ANISOVICH 1998B
SPU 41 419 The Lightest Scalar Glueball
 BERTIN 1998B
PL B434 180 Study of the Isovector Scalar Mesons in the Channel ${{\overline{\mathit p}}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit K}^{\pm}}{{\mathit K}_S^0}$ ${{\mathit \pi}^{\mp}}$ at Rest with Initial Angular Momentum State
 TORNQVIST 1996
PRL 76 1575 Resurrection of the ${{\mathit \sigma}}$ Meson
 JANSSEN 1995
PR D52 2690 On the Structure of the Scalar Mesons ${{\mathit f}_{{0}}{(975)}}$ and ${{\mathit a}_{{0}}{(980)}}$
 AMSLER 1994C
PL B327 425 Study of ${{\overline{\mathit p}}}{{\mathit p}}$ Annihilation at Rest into ${{\mathit \omega}}{{\mathit \eta}}{{\mathit \pi}^{0}}$
 AMSLER 1992
PL B291 347 Proton Antiproton Annihilation into ${{\mathit \eta}}{{\mathit \eta}}{{\mathit \pi}}$: Observation of a Scalar Resonance Decaying into ${{\mathit \eta}}{{\mathit \eta}}$
 ARMSTRONG 1991B
ZPHY C52 389 Study of the ${{\mathit \eta}}{{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ System Centrally Produced in the Reaction ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit p}_{{f}}}$( ${{\mathit \eta}}{{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$) ${{\mathit p}_{{s}}}$ at 300 ${\mathrm {GeV/}}\mathit c$
 EVANGELISTA 1981
NP B178 197 Results from a ${{\mathit \pi}^{-}}{{\mathit p}}$ Slow Proton Trigger Experiment at 12 ${\mathrm {GeV/}}\mathit c$
 GURTU 1979
NP B151 181 Production and Decay Properties of the ${{\mathit D}{(1285)}}$ Meson in ${{\mathit K}^{-}}{{\mathit p}}$ Interactions at 4.2 ${\mathrm {GeV/}}\mathit c$
 CONFORTO 1978
LNC 23 419 Study of the ${{\mathit \delta}{(970)}}$ Meson
 CORDEN 1978
NP B144 253 Observation of the ${{\mathit D}}$, ${{\mathit E}}$ and ${{\mathit \delta}}$ Mesons in ${{\mathit \pi}^{-}}{{\mathit p}}$ Interactions at 12 and 15 ${\mathrm {GeV/}}\mathit c$
 GRASSLER 1977
NP B121 189 Production and Decay of the ${{\mathit \delta}{(970)}}$ and ${{\mathit D}{(1285)}}$ Mesons in ${{\mathit \pi}^{\pm}}{{\mathit p}}$ Interactions at 16 ${\mathrm {GeV/}}\mathit c$
 FLATTE 1976
PL 63B 224 Coupled-Channel Analysis of the ${{\mathit \pi}}{{\mathit \eta}}$ and ${{\mathit K}}{{\overline{\mathit K}}}$ Systems near ${{\mathit K}}{{\overline{\mathit K}}}$ Threshold
 WELLS 1975
NP B101 333 $\delta$ Production in $3.1 - 3.6$ ${\mathrm {GeV/}}\mathit c$ ${{\mathit K}^{-}}{{\mathit p}}$ Interactions
 DEFOIX 1972
NP B44 125 Evidence for Decays of the ${{\mathit D}}$- and ${{\mathit E}}$-Mesons into ${{\mathit \sigma}}{{\mathit \pi}}$ in ${{\overline{\mathit p}}}{{\mathit p}}$ Annihilation at 700 ${\mathrm {MeV}}/\mathit c$
 CAMPBELL 1969
PRL 22 1204 Study of ${{\mathit D}^{0}}$ $\rightarrow$ ${{\mathit \pi}^{\pm}}{{\mathit \delta}^{\mp}}$ and ${{\mathit \delta}^{\mp}}$ $\rightarrow$ ${{\mathit \pi}^{\mp}}{{\mathit \eta}}$ in the Reaction ${{\mathit \pi}^{+}}$ ${{\mathit d}}$ $\rightarrow$ ${{\mathit p}}$ $_{spect.}$ ${{\mathit p}}{{\mathit D}^{0}}$ at 2.7 ${\mathrm {GeV/}}\mathit c$
 MILLER 1969B
PL 29B 255 Production of a ${{\mathit \pi}}-{{\mathit \eta}}$ Resonance at a Mass of 0.98 GeV in ${{\mathit K}^{-}}$ ${{\mathit n}}$ $\rightarrow$ ${{\mathit \Lambda}}{{\mathit \pi}^{-}}{{\mathit \eta}}$ at 4.5 ${\mathrm {GeV/}}\mathit c$
 AMMAR 1968
PRL 21 1832 Boson Resonance of Mass 980 MeV Decaying into ${{\mathit \pi}}{{\mathit \eta}}$