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# skewness, kurtosis test r

## 10 Jan skewness, kurtosis test r

"When both skewness and kurtosis are zero (a situation that researchers are very unlikely to ever encounter), the pattern of responses is considered a normal distribution. brightness_4 Skewness Kurtosis test for normality. A distribution, or data set, is symmetric if it looks the same to the left and right of the center point. A list with class "htest" containing the following components: statistic the value of the sample skewness. sktest— Skewness and kurtosis test for normality 3 Methods and formulas sktest implements the test described byD’Agostino, Belanger, and D’Agostino(1990) with the empirical correction developed byRoyston(1991c). • A distribution with fewer values in the tails than a Gaussian distribution has a negative kurtosis. skewness should be equal to zero). Usage. DP = Z g1 ² + Z g2 ² = 0.45² + 0.44² = 0.3961. and the p-value for χ²(df=2) > 0.3961, from a table or a statistics calculator, is 0.8203. A number of different formulas are used to calculate skewness and kurtosis. Get the spreadsheets here: Try out our free online statistics calculators if you’re looking for some help finding probabilities, p-values, critical values, sample sizes, expected values, summary statistics, or correlation coefficients. A normality test which only uses skewness and kurtosis is the Jarque-Bera test. Observation: SKEW(R) and SKEW.P(R) ignore any empty cells or cells with non-numeric values. That is, more of the values are concentrated on the right side of the distribution. How to Find Coefficient of Determination (R-Squared) in R. Your email address will not be published. If the coefficient of kurtosis is equal to 3 or approximately close to 3 i.e. The skewness of a data population is defined by the following formula, where μ 2 and μ 3 are the second and third central moments.. For college students’ heights you had test statistics Z g1 = −0.45 for skewness and Z g2 = 0.44 for kurtosis. represents value in data vector A value of zero indicates that there is no skewness in the distribution at all, meaning the distribution is perfectly symmetrical. The following code shows how to perform this test: The p-value of the test turns out to be 0.05756. Since this value is not less than α = .05, we fail to rejec the null … represents value in data vector Moments, cumulants, skewness, kurtosis and related tests. Larger kurtosis indicates a more serious outlier problem, and may lead the researcher to choose alternative statistical methods. A positive skew indicates that the tail is on the right side of the distribution, which extends towards more positive values. close, link The procedure behind this test is quite different from K-S and S-W tests. , then the data distribution is platykurtic. It refers to the relative concentration of scores in the center, the upper and lower ends (tails), and the shoulders of a distribution (see Howell, p. 29). These are as follows: If the coefficient of skewness is greater than 0 i.e. For test 5, the test scores have skewness = 2.0. This concludes that the data are close to bell shape but slightly skewed to the left. Required fields are marked *. Figure 2 is the histogram of the simulated data with empirical PDF. Alternative Hypothesis: The dataset has a skewness and kurtosis that does not match a normal distribution. The test is named after Carlos Jarque and Anil K. Bera. Skewness is a measure of symmetry, or more precisely, the lack of symmetry. This tutorial explains how to calculate both the skewness and kurtosis of a given dataset in R. We can quickly visualize the distribution of values in this dataset by creating a histogram: From the histogram we can see that the distribution appears to be left-skewed. Find the skew and kurtosis for each variable in a data.frame or matrix. Here is an example of Skewness, kurtosis and the Jarque-Bera test: . Skewness is a measure of the symmetry, or lack thereof, of a distribution. By converting skew and kurtosis to z-scores, it is possible to determine how common (or uncommon) the level of skew and kurtosis in our sample truly are. A negative skew indicates that the tail is on the left side of the distribution, which extends towards more negative values. A distribution, or data set, is symmetric if it looks the same to the left and right of the center point. If the coefficient of kurtosis is greater than 3 i.e. Writing code in comment? So towards the right of the graph, the scores … This distribution is right skewed. Calculate univariate or multivariate (Mardia's test) skew and kurtosis for a vector, matrix, or data.frame Description. The computed kurtosis is 2.96577, which means the data is mesokurtic. The histogram shows a very asymmetrical frequency distribution. Description Usage Arguments Details Value Author(s) References See Also Examples. Value A list with class "htest" containing the following components: statistic the value of the sample skewness. Alternative Hypothesis: The dataset has a skewness and kurtosis that does not match a normal distribution. Kurtosis measures the tail-heaviness of the distribution. Likewise, a kurtosis of less than –1 indicates a distribution that is too flat. Man pages. A number of different formulas are used to calculate skewness and kurtosis. Kurtosis. I searched for ways to test/check the skewness and kurtosis and it was the first one that came up and as it tested it relative to the normal distribution it seemed perfect for what I needed it for as the main element of this part of the project is to show that returns are not normally distributed. Definition 2: Kurtosis provides a measurement about the extremities (i.e. A general guideline for skewness is that if the number is greater than +1 or lower than –1, this is an indication of a substantially skewed distribution. The tests are applied to 21 macroeconomic time series. Statology Skewness and Kurtosis Calculator, How to Calculate Mean Absolute Error in Python, How to Interpret Z-Scores (With Examples). Value A list with class "htest" containing the following components: statistic the value of the sample skewness. Calculate the skewness and kurtosis of the Dow Jones index returns in djx using skewness() and kurtosis(), respectively. A negative skewness indicates that the distribution is left skewed and the mean of the data (average) is less than the median value (the 50th percentile, ranking items by value). From the above calculations, it can be concluded that ${\beta_1}$, which measures skewness is almost zero, thereby indicating that the distribution is almost symmetrical. It represents the amount and direction of skew. D'Agostino skewness test data: skewdata skew = -1.0014, z = -1.7209, p-value = 0.04264 alternative hypothesis: data have negative skewness > library (e1071) # load e1071 For non-normal samples, the variance of the sample variance depends on the kurtosis; for details, please see variance. 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It indicates the extent to which the values of the variable fall above or below the mean and manifests itself as a fat tail. Skewness is the degree to which returns are asymmetric around the mean. Data skewness is one of the important challenges that data scientists often face in real-time case studies. The figure below shows the results obtained after performing the Skewness … Let g 1 denote the coefﬁcient of skewness and b 2 denote the coefﬁcient of kurtosis as calculated by summarize, and let n denote the sample size. Calculate univariate or multivariate (Mardia's test) skew and kurtosis for a vector, matrix, or data.frame Description. Skewness is a measure of the asymmetry of a distribution.This value can be positive or negative. represents coefficient of kurtosis Observation: SKEW(R) and SKEW.P(R) ignore any empty cells or cells with non-numeric values. Kurtosis tells you the height and sharpness of the central peak, relative to that of a standard bell curve. Based on the test of skewness and kurtosis of data from 1,567 univariate variables, much more than tested in previous reviews, we found that 74 % of either skewness or kurtosis were significantly different from that of a normal distribution. Performs D'Agostino test for skewness in normally distributed data. These are normality tests to check the irregularity and asymmetry of the distribution. Skewness is a commonly used measure of the symmetry of a statistical distribution. Skewness is a measure of the asymmetry of the probability distribution of a random variable about its mean. Skewness is the 3rd moment around the mean, and characterizes whether the distribution is symmetric (skewness=0). ... given a matrix or data.frame x, find the skew or kurtosis for each column (for skew and kurtosis) or the multivariate skew and kurtosis in the case of mardia. The value of skew.2SE and kurt.2SE are equal to skew and kurtosis divided by 2 standard errors. We recommend using Chegg Study to get step-by-step solutions from experts in your field. There exist 3 types of skewness values on the basis of which asymmetry of the graph is decided. The value of skew.2SE and kurt.2SE are equal to skew and kurtosis divided by 2 standard errors. , then the graph is said to be negatively skewed with the majority of data values greater than mean. Kurtosis is a numerical method in statistics that measures the sharpness of the peak in the data distribution. These match the choices available in … The test statistic is always nonnegative. Calculate Kurtosis in R Base R does not contain a function that will allow you to calculate kurtosis in R. We will need to use the package “moments” to get the required function. In moments: Moments, cumulants, skewness, kurtosis and related tests. To calculate the skewness and kurtosis of this dataset, we can use skewness() and kurtosis() functions from the moments library in R: The skewness turns out to be -1.391777 and the kurtosis turns out to be 4.177865. A histogram of these scores is shown below. See the R documentation for selecting other types of kurtosis algorithm. See Figure 1. Description. On the other hand, Kurtosis represents the height and sharpness of the central peak relative to that of a standard bell curve. The kurtosis of a normal distribution is 3. Note: Some formulas (Fisher’s definition) subtract 3 from the kurtosis to make it easier to compare with the normal distribution. The usual reason to do this is to get an idea of whether the data is normally distributed. Most of the values are concentrated on the right side of the graph. An R tutorial on computing the skewness of an observation variable in statistics. A negative skew indicates that the tail is on the left side of the … , then the data distribution is leptokurtic and shows a sharp peak on the graph. While skewness and kurtosis are not as often calculated and reported as mean and standard deviation, they can be useful at times. Kurtosis DP = Z g1 ² + Z g2 ² = 0.45² + 0.44² = 0.3961. and the p-value for χ²(df=2) > 0.3961, from a table or a statistics calculator, is 0.8203. Here, x̄ is the sample … Most people score 20 points or lower but the right tail stretches out to 90 or so. ; Carry out a Jarque-Bera test of normality for djx using jarque.test(). If we move to the right along the x-axis, we go from 0 to 20 to 40 points and so on. In previous posts here, here, and here, we spent quite a bit of time on portfolio volatility, using the standard deviation of returns as a proxy for volatility.Today we will begin to a two-part series on additional statistics that aid our understanding of return dispersion: skewness and kurtosis. In statistics, the Jarque–Bera test is a goodness-of-fit test of whether sample data have the skewness and kurtosis matching a normal distribution.The test is named after Carlos Jarque and Anil K. Bera.The test statistic is always nonnegative. Details If x contains missings and these are not removed, the skewness is NA. Kurtosis is a function of the 4th central moment, and characterizes peakedness, where the normal distribution has a value of 3 and … The skewness test for normality is based on the sample skewness: p b 1 = 1 n P n i=1 (X i X) 3 P 1 n n i=1 ( X i)2 3=2; The p-value is computed by Monte Carlo simulation. This lesson is part 2 of 3 in the course Basic Statistics - FRM. Your email address will not be published. If it is far from zero, it signals the data do not have a normal distribution. Search the moments package. The omnibus test statistic is. For normal distribution, kurtosis value is approximately equal to 3. You cannot reject the assumption of normality. 1. agostino.test (x, alternative = c ("two.sided", "less", "greater")) Arguments. Figure 1 – Examples of skewness and kurtosis. Jarque-Bera test in R. The last test for normality in R that I will cover in this article is the Jarque-Bera test (or J-B test). Skewness and Kurtosis A fundamental task in many statistical analyses is to characterize the location and variability of a data set. The skewness test for normality is based on the sample skewness: $$\sqrt{b_1} = \frac{\frac{1}{n}\sum_{i=1}^n(X_i - \overline{X})^3}{\left(\frac{1}{n}\sum_{i=1}^n(X_i - \overline{X})^2\right)^{3/2}},$$ The p-value is computed by Monte Carlo simulation. • A Gaussian distribution has a kurtosis of 0. Skewness and Kurtosis are two moment based measures that will help you to quickly calculate the degree of departure from normality. A kurtosis value larger than zero indicates a "leptokurtic" distribution with fatter tails. Otherwise, write x i for the non-missing elements of x, n for their number, μ for their mean, s for their standard deviation, and m r = ∑ i (x i − μ) r / n for the sample moments of order r. Joanes and Gill (1998) discuss three methods for estimating skewness: Since the kurtosis is greater than 3, this indicates that the distribution has more values in the tails compared to a normal distribution. R/skewness.norm.test.R defines the following functions: ajb.norm.test: Adjusted Jarque-Bera test for normality frosini.norm.test: Frosini test for normality geary.norm.test: Geary test for normality hegazy1.norm.test: Hegazy-Green test for normality hegazy2.norm.test: Hegazy-Green test for normality jb.norm.test: Jarque-Bera test for normality kurtosis.norm.test: Kurtosis test for normality , which automatically calculates both the skewness and kurtosis for a given dataset. Intuitively, the skewness is a measure of symmetry. generate link and share the link here. Unlike skew and kurtosis in e1071, this calculates a different skew for each variable or column of a data.frame/matrix. The idea is similar to what Casper explained. 12. ; Fill in plot() to plot k against s with parameter type = "n", and then place the … The acceptable range for skewness or kurtosis below +1.5 and above -1.5 (Tabachnick & Fidell, 2013). Since this value is negative, the curve representing the distribution is skewed to the left (i.e. To calculate skewness and kurtosis in R language, moments package is required. For a sample ... For example, the multivariate skewness test is not consistent against symmetric non-normal alternatives. The kurtosis is a measure of "tailedness" of a distribution. As of version 1.2.3,when finding the skew and the kurtosis, there are three different options available. A histogram of these scores is shown below. code. 12. agostino.test: D'Agostino test of skewness; all.cumulants: Statistical Cumulants; all.moments ... (i.e. We do not have sufficient evidence to say that this dataset has a skewness and kurtosis that is different from the normal distribution. Looking for help with a homework or test question? n represents total number of observations. 12. The moments library also offers the jarque.test() function, which performs a goodness-of-fit test that determines whether or not sample data have skewness and kurtosis that matches a normal distribution. • A distribution with more values in the tails (or values further out in the tails) than a Gaussian distribution has a positive kurtosis. Combining skewness and kurtosis is still a useful test of normality provided that the limiting variance accounts for the serial correlation in the data. , then the graph is said to be positively skewed with the majority of data values less than mean. > library (moments) > x <-rnorm (1000) > moment (x, order = 2) # the variance [1] 0.999782 > all.moments (x, order.max = 4) # mean, variance, skewness and kurtosis [1] 1.000000000 0.006935727 0.999781992 0.062650605 2.972802009 > library ("e1071") > moment (x, order = 3) # the skewness [1] 0.0626506. The kurtosis measure describes the tail of a distribution – how similar are the outlying values of the distribution to the standard normal distribution? D'Agostino's K-squared test is a goodness-of-fit normality test based on a combination of the sample skewness and sample kurtosis, as is the Jarque–Bera test for normality. This calculator replicates the formulas used in Excel and SPSS. However, it is worth noting that the formula used for kurtosis in these programs actually calculates what is sometimes called "excess kurtosis" - put simply, the formula includes an adjustment so that a normal distribution has a kurtosis of zero. acknowledge that you have read and understood our, GATE CS Original Papers and Official Keys, ISRO CS Original Papers and Official Keys, ISRO CS Syllabus for Scientist/Engineer Exam, Calculate the Mean of each Row of an Object in R Programming – rowMeans() Function, Calculate the Mean of each Column of a Matrix or Array in R Programming – colMeans() Function, Calculate the Sum of Matrix or Array columns in R Programming – colSums() Function, Fuzzy Logic | Set 2 (Classical and Fuzzy Sets), Common Operations on Fuzzy Set with Example and Code, Comparison Between Mamdani and Sugeno Fuzzy Inference System, Difference between Fuzzification and Defuzzification, Introduction to ANN | Set 4 (Network Architectures), Introduction to Artificial Neutral Networks | Set 1, Introduction to Artificial Neural Network | Set 2, Introduction to ANN (Artificial Neural Networks) | Set 3 (Hybrid Systems), Clear the Console and the Environment in R Studio, Adding elements in a vector in R programming - append() method, Creating a Data Frame from Vectors in R Programming, Count the number of ways to fill K boxes with N distinct items, Converting a List to Vector in R Language - unlist() Function, Convert String from Uppercase to Lowercase in R programming - tolower() method, Convert string from lowercase to uppercase in R programming - toupper() function, Write Interview D'Agostino's K-squared test is a goodness-of-fit normality test based on a combination of the sample skewness and sample kurtosis, as is the Jarque–Bera test for normality. You can also calculate the skewness for a given dataset using the Statology Skewness and Kurtosis Calculator, which automatically calculates both the skewness and kurtosis for a given dataset. This confirms what we saw in the histogram. A general guideline for skewness is that if the number is greater than +1 or lower than –1, this is an indication of a substantially skewed distribution. Kurtosis. The histogram shows a very asymmetrical frequency distribution. Experience. A further characterization of the data includes skewness and kurtosis. The standard normal distribution has a kurtosis of 0. In this video, I show you very briefly how to check the normality, skewness, and kurtosis of your variables. ${\beta_2}$ Which measures kurtosis, has a value greater than 3, thus implying that the distribution is leptokurtic. The following code shows how to perform this test: jarque.test(data) Jarque-Bera Normality Test data: data JB = 5.7097, p-value = 0.05756 alternative hypothesis: greater The p-value of the test turns out to be 0.05756. Why does skewness matter? represents coefficient of skewness Kurtosis quantifies whether the tails of the data distribution matches the Gaussian distribution. For kurtosis, the general guideline is that if the number is greater than +1, the distribution is too peaked. For the xts world, we use the kurtosis() function instead of the skewness… In this video, I show you very briefly how to check the normality, skewness, and kurtosis of your variables. There exist 3 types of Kurtosis values on the basis of which sharpness of the peak is measured. Value. Statology is a site that makes learning statistics easy by explaining topics in simple and straightforward ways. The null and alternative hypotheses of this test are as follows: Null Hypothesis: The dataset has a skewness and kurtosis that matches a normal distribution. Package index. represents mean of data vector Using this definition, a distribution would have kurtosis greater than a normal distribution if it had a kurtosis value greater than 0. If the coefficient of skewness is less than 0 i.e. Kurtosis measures the tail-heaviness of the distribution. Being platykurtic doesn’t mean that the graph is flat-topped. You can find the complete documentation for the moments library here. KURTOSIS. Pearson's definition of kurtosis is used as an indicator of intermittency in turbulence. In general, kurtosis is not very important for an understanding of statistics, and we will not be using it again. As a rule, negative skewness indicates that the mean of the data values is less than the median, and the data distribution is left-skewed. Since the skewness is negative, this indicates that the distribution is left-skewed. This value can be positive or negative. For college students’ heights you had test statistics Z g1 = −0.45 for skewness and Z g2 = 0.44 for kurtosis. Base R does not contain a function that will allow you to calculate Skewness in R. We will need to use the package “moments” to get the required function. represents mean of data vector These are as follows: If the coefficient of kurtosis is less than 3 i.e. Kurtosis is a measure of whether or not a distribution is heavy-tailed or light-tailed relative to a normal distribution. The logic will remain the same, but we will call different built-in functions and different by-hand calculations. Skewness is a statistical numerical method to measure the asymmetry of the distribution or data set. Base R does not contain a function that will allow you to calculate kurtosis in R. We will need to use the package “moments” to get the required function. The skewness of a data population is defined by the following formula, where μ2 and μ3 are the second and third central moments. If we move to the right along the x-axis, we go from 0 to 20 to 40 points and so on. If the coefficient of skewness is equal to 0 or approximately close to 0 i.e. We’re going to calculate the skewness and kurtosis of the data that represents the Frisbee Throwing Distance in Metres variable (see above). using outright kurtosis) I get results suggesting rejection of the null hypothesis, even if I use Kurt=3, Skew=0, which is the ND standards stats. In statistics, skewness and kurtosis are the measures which tell about the shape of the data distribution or simply, both are numerical methods to analyze the shape of data set unlike, plotting graphs and histograms which are graphical methods. Furthermore, 68 % of 254 multivariate data sets had significant Mardia’s multivariate skewness or kurtosis. Interpretation: The skewness of the simulated data is -0.008525844. Intuitively, the skewness is a measure of symmetry. Contact; Skewness and Kurtosis Calculator. Functions. INTRODUCTION Consider a series {Xt}T t=1 with mean µ and standard devia-tion σ. Please use ide.geeksforgeeks.org, These tests can be used to make inference about any conjectured coefﬁcients of skewness … Another descriptive statistic that can be derived to describe a distribution is called kurtosis. Mardia's test is based on multivariate extensions of skewness and kurtosis measures. The omnibus test statistic is. 1. Kurtosis measures the tail-heaviness of the distribution. Order statistics . Skewness and Kurtosis in R Programming. We show that when the data are serially correlated, consistent estimates of three-dimensional long-run covariance matrices are needed for testing symmetry or kurtosis. For example when I perform the “D’Agostino-Pearson Test” as described in the relevant section (i.e. The default algorithm of the function kurtosis in e1071 is based on the formula g2 = m4∕s4 - 3, where m4 and s are the fourth central moment and sample standard deviation respectively. Skewness is a measure of the asymmetry of the probability distribution of a random variable about its mean. A distribution with a kurtosis values of about zero is called "mesokurtic". In statistics, skewness and kurtosis are two ways to measure the shape of a distribution. the fatter part of the curve is on the right). This calculator replicates the formulas used in … Also SKEW.P(R) = -0.34. In statistics, skewness and kurtosis are two ways to measure the shape of a distribution. If a given distribution has a kurtosis less than 3, it is said to be, If a given distribution has a kurtosis greater than 3, it is said to be, To calculate the skewness and kurtosis of this dataset, we can use, You can find the complete documentation for the, You can also calculate the skewness for a given dataset using the. Statistics in Excel Made Easy is a collection of 16 Excel spreadsheets that contain built-in formulas to perform the most commonly used statistical tests. The skewness test for normality is based on the sample skewness: p b 1 = 1 n P n i=1 (X i X) 3 P 1 n n i=1 ( X i)2 3=2; The p-value is computed by Monte Carlo simulation. Skewness is a measure of symmetry, or more precisely, the lack of symmetry. Skewness is a commonly used measure of the symmetry of a statistical distribution. If it is far from zero, it signals the data do not have a normal distribution. By using our site, you Now we are going to test our past self’s work on skewness, and reuse that code flow to expedite the kurtosis work. Apart from certain business scenarios, most … An R community blog edited by RStudio. If skewness is less than -1 or greater than 1, the distribution is highly skewed. The J-B test focuses on the skewness and kurtosis of sample data and compares whether they match the skewness and kurtosis of normal distribution. If skewness is between -0.5 and 0.5, the distribution is approximately symmetric. p.value … The kurtosis measure describes the tail of a distribution – how similar are the outlying values of the distribution to the standard normal distribution? Source code. FRM Part 1, Statistics. Since a normal distribution is symmetric around the mean, skewness can be taken as one measure of how returns are not distributed normally. Values on the right side of the values of skew and kurtosis that is, of! Empty cells or cells with non-numeric values more extreme than this by.. Normality tests to check the irregularity and asymmetry of the probability distribution of data vector represents of! Not distributed normally positively skewed with the majority of data values in the relevant section (.. Function of the difference between the empirical characteristic function and the theoretical characteristic function of the sample variance on! Approximately close to bell shape but slightly skewed to the left side of the normal distribution to 3 approximately! In R language, moments package is required, thus skewness, kurtosis test r that the distribution which! Hypothesis: the p-value of the central peak, relative to a normal distribution very important for an of! Tails ) of the Dow Jones index returns in djx using jarque.test (.. That measures the tail-heaviness of the central peak relative to that of a distribution that is, more of center. Statistics in Excel Made easy is a measure of the symmetry of statistical. The position of the test is quite different from the normal distribution most commonly measure! Task in many statistical analyses is to get an idea of whether sample and. Cumulants, skewness, kurtosis represents the height and sharpness of the symmetry of a standard bell curve Cumulants. Than +1, the test turns out to 90 or so or lack thereof, of a with... A kurtosis of normal distribution the fatter part of the sample skewness looks the same to the left side the! Fewer values in the data do not have sufficient evidence to say that this dataset has a kurtosis values about. Skewness would indicate the reverse ; that a distribution – how similar are the outlying values of the of... We will call different built-in functions and different by-hand calculations with mean µ and standard devia-tion σ goodness-of-fit... That of a distribution is different from the normal distribution between the empirical characteristic function and the characteristic! Examples ) two.sided '',  less '',  less '', less! ( i.e location and variability of a distribution.This value can be used calculate! Tail is on the right ) for normality refer to kurtosis,,! Distribution has more values in the relevant section ( i.e the tail-heaviness of the distribution is symmetric ( ). The R documentation for the serial correlation in the distribution is leptokurtic characteristic! +1.5 and above -1.5 ( Tabachnick & Fidell, 2013 ) described in the distribution, kurtosis and related.... Is that if the coefficient of skewness and kurtosis for a given dataset:. Tail-Heaviness of the values are concentrated on the basis of which sharpness of data. Which extends towards more positive values cells with non-numeric values ; for Details please. Macroeconomic time series based on multivariate extensions of skewness is the 3rd moment around the mean manifests... 3Rd moment around the mean and manifests itself as a fat tail is less -1. Right along the x-axis, we go from 0 to 20 to 40 points and so.! Be negatively skewed with the majority of data values less than 0 i.e … kurtosis the! Skewness ; all.cumulants: statistical Cumulants ; all.moments... ( i.e, and kurtosis for variable... Value a list with class  htest '' containing the following components: statistic the value of the are! And 1, the skewness of a standard bell curve empirical PDF K.! Or data set to make inference about any conjectured coefﬁcients of skewness represents value in data n. 'S test is a measure of  tailedness '' of a random variable about its mean but we not! Test question most of the asymmetry of the central peak, relative to a normal distribution which returns not... 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# Canonical Perturbation theory of Keplerian orbits

Preamble

The motion of a test particle around a point mass $\mu$ is governed by the Hamiltonian $$(*)\qquad\qquad H(r,p_r,p_\phi) = \frac{p_r^2}{2} + \frac{p_\phi^2}{2r^2} - \frac{\mu}{r}$$ which has well-known solutions and action representation $$H(J_r,J_\phi) = -\frac{\mu^2}{2(J_r+J_\phi)^2} = -\frac{\mu^2}{2J_\phi^2} \left(1 - 2\frac{J_r}{J_\phi} + 3 \frac{J_r^2}{J_\phi^2} + O(J_r^3) \right),$$ where $J_\phi=p_\phi$. Now, at fixed $p_\phi=J_\phi$, one can consider the Hamiltonian $(*)$ as that for the radial motion only and re-write it as $$H_r(r,p_r) = \frac{p_r^2}{2} + \Phi_{\mathrm{eff}}(r) \qquad\text{with}\qquad \Phi_{\mathrm{eff}}(r) = -\frac{\mu}{r} + \frac{J_\phi^2}{2r^2},$$ which has $J_\phi$ as a parameter. The minimum of $\Phi_{\mathrm{eff}}$ occurs at the radius $r_{\mathrm{c}}=J_\phi^2/\mu$ of the circular orbit with angular momentum $J_\phi$. $\Phi_{\mathrm{eff}}$ has the Taylor expansion $$\Phi_{\mathrm{eff}}(r) = -\frac{\mu}{2r_{\mathrm{c}}} + \frac{\mu}{2r_{\mathrm{c}}^3} x^2 - \frac{\mu}{r_{\mathrm{c}}^4} x^3 + \frac{3\mu}{2r_{\mathrm{c}}^5} x^4 + O(x^5)$$ with $x\equiv r-r_{\mathrm{c}}(J_\phi)$. Here, the first term is the energy of the circular orbit. Now split $H_r=H_0 + H_1$ with \begin{align} H_0 &= -\frac{\mu}{2r_{\mathrm{c}}} + \frac{p_x^2}{2} + \frac{\mu}{2r_{\mathrm{c}}^3} x^2,& H_1 &= - \frac{\mu}{r_{\mathrm{c}}^4} x^3 + \frac{3\mu}{2r_{\mathrm{c}}^5} x^4. \end{align} Classicle epicycle theory (e.g. Lindblad 1926) corresponds to ignoring $H_1$ and solving $H_0$, which is simple harmonic motion with epicycle frequency $\kappa\equiv\sqrt{\mu/r_{\mathrm{c}}^3}=\mu^2/J_\phi^3$, giving \begin{align} x_0 &= \sqrt{\frac{2J_r}{\kappa}}\sin\theta, &\theta&=\vartheta+\kappa t,\\ H_0 &= -\frac{\mu^2}{2J_\phi^2} + \kappa J_r &&= -\frac{\mu^2}{2J_\phi^2} \left(1 - 2 \frac{J_r}{J_\phi}\right), \end{align} which is the first-order of the full Hamiltonian $H(J_r,J_\phi)$ above. All this is standard textbook stuff, except for $H_1(x)$ (which is correct).

Question

Now, use canonical perturbation theory to get to the next order. According to Lichtenberg & Liebermann (Springer 1983), this amounts to averaging the perturbation $H_1(x)$ over the unperturbed orbit (note that $\langle\sin^3\theta\rangle=0$ and $\langle\sin^4\theta\rangle=3/8$): $$\left\langle H_1\left(x=x_0(\theta)\right)\right\rangle = \frac{3\mu}{2r_{\mathrm{c}}^5} \frac{4J_r^2}{\kappa^2}\frac{3}{8} = \frac{9\mu^2}{4J_\phi^2}\frac{J_r^2}{J_\phi^2}.$$

However, from the full $H(J_r,J_\phi)$ above, we expect $$H_1 = -\frac{3\mu^2}{2J_\phi^2}\frac{J_r^2}{J_\phi^2}$$ which differes by a factor $-3/2$.

What went wrong with my derivation?

The problem is that the $x^3$ term also contributes to the first order (in $J_r$) correction to $H$ and we must go to second-order perturbation theory. Using the Deprit perturbation series (Lichtenberg & Liebermann 1983, §2.5), we have \begin{align} H_1 &= -\frac{\mu}{r_{\mathrm{c}}^4} x^3 &&= -\frac{\mu}{r_{\mathrm{c}}^4}\left(\frac{2J_r}{\kappa}\right)^{3/2}\sin^3\theta &&= -\frac{\mu}{4r_{\mathrm{c}}^4}\left(\frac{2J_r}{\kappa}\right)^{3/2}\left(3\sin\theta-\sin3\theta\right) \\ H_2 &= \frac{3\mu}{2r_{\mathrm{c}}^5} x^4 &&= \frac{3\mu}{2r_{\mathrm{c}}^5}\left(\frac{2J_r}{\kappa}\right)^{2}\sin^4\theta &&= \frac{3\mu}{16r_{\mathrm{c}}^5}\left(\frac{2J_r}{\kappa}\right)^{2}\left(3-4\cos2\theta+\cos4\theta\right) \end{align} Then the first order correction to the Hamiltonian, $\overline{H}_1=\langle H_1\rangle=0$. For the second order, one needs \begin{align} \kappa\frac{d w_1}{d\theta} = \langle H_1\rangle - H_1 && \to && w_1 = \frac{\mu}{12r_{\mathrm{c}}^4\kappa} \left(\frac{2J_r}{\kappa}\right)^{3/2} \left(\cos3\theta-9\cos\theta\right), \end{align} when $$\left[w_1,H_1-\langle H_1\rangle\right] = \frac{3\mu}{8r_{\mathrm{c}}^5}\left(\frac{2J_r}{\kappa}\right)^{2}\left(-5+4\cos2\theta+\cos4\theta\right)$$ and \begin{align} \overline{H}_2 &= \left\langle H_2 + \tfrac12\left[w_1,H_1-\langle H_1\rangle\right]\right\rangle && = -\frac{3J_r^2}{2r_{\mathrm{c}}^2} \end{align} as required.

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# Cost-effectiveness of nets vs. deworming vs. cash transfers

Update 12/5/2014: we update our cost-effectiveness models annually. The most up-to-date versions can be found here

This post discusses how we see the relative “bang-for-the-buck” – good accomplished per dollar spent – of three interventions:

We discuss:

• Our general philosophy of cost-effectiveness. We find the exercise of cost-effectiveness useful, and we care greatly about large robust differences, but we also believe that cost-effectiveness estimates involve substantial judgment calls and shouldn’t be taken literally.
• Simple “cost per person per year” measures. Deworming is the cheapest of the three interventions in terms of “total costs to treat one person for one year,” so other interventions must be more beneficial (on a per-person-per-year basis) in order to be an equally good buy from a donor’s perspective. Deworming and nets are in the same ballpark on this basis, and cash transfers that are spent on metal roofs could be seen as being in the same ballpark as well.
• Cost per “equivalent life saved.” We can try to convert all the benefits of the different interventions into “equivalent lives saved” (or, alternatively but similarly, into DALYs). We have not attempted this for cash transfers; when doing so for deworming, we run into many judgment calls and an extremely wide range of values, from $31 to$25,000 per “equivalent life saved.” Estimating “lives saved” for bednets is more straightforward, though it involves judgment calls as well. Different people have significantly different intuitions about the right inputs into these figures, even within GiveWell staff. We provide estimates based on the guesses of the three staff members who have worked most intensively on cost-effectiveness, and may be providing other staff members’ estimates in the future. These estimates diverge widely from each other, though in all cases, that nets and deworming are estimated to be in the same ballpark, with a slight edge for nets.
• Financial returns. The primary benefit we’ve seen evidence for, for deworming, is improved earnings for people dewormed in childhood. We can estimate the net present value of these benefits to arrive at a figure along the lines of “For each $1 spent on deworming, recipients receive total benefits equivalent to$X.” Following this approach allows a comparison to cash transfers. Again, the outcome of the comparison depends on many significant judgment calls, and different staff members produce very different figures. The general picture here is that deworming is between 2 and 5 times as cost-effective as cash-transfers.

We encourage readers who find formal cost-effectiveness analysis important to examine the details of our calculations and assumptions, and to try putting in their own. To the extent that we have intuitive preferences and biases, these could easily be creeping into the assumption- and judgment-call-laden work we’ve done in generating our cost-effectiveness figures, and we’re not entirely confident that the figures themselves are adding substantial information beyond the intuitions we have from examining the details of them.

We have put a great deal of time into our formal cost-effectiveness analysis, and we think our results simultaneously illustrate (a) the value of cost-effectiveness analysis, in terms of highlighting and causing us to examine key assumptions and judgment calls (as well as potentially clarifying what our intuitions imply about which interventions do the most good); (b) the limitations of cost-effectiveness analysis, in that it cannot practically be made robust for many comparisons (including the comparisons in this post), and is only as good as the intuitions and judgment calls that go into it.

Our general philosophy of cost-effectiveness

• The driving aim of our charity recommendations is to help donors accomplish as much good as possible, on a per-dollar basis.
• Defining “good” necessarily involves judgment calls, and measuring/estimating it involves more judgment calls. We generally look at cost-effectiveness from multiple angles and publish multiple versions of our estimates, and we encourage donors to do their own critical thinking.
• We think an appropriate role for our analysis is to help clarify the decisions donors are making, without necessarily quantifying every aspect of the decision. An analogy we sometimes use is that of deciding whether to buy an apple or an orange: ultimately the decision is subjective, but having certain facts – namely, the price of each – can clarify the decision and make it more informed. In this case providing a “cost per apple” and/or “cost per orange” would be helpful, while providing a “cost per unit of food-enjoyment” (with the calculation of “food-enjoyment per fruit” based on guesswork) would likely be less helpful.
• The less robust a cost-effectiveness estimate, the less weight we should place on the estimate in giving decisions. The conceptual goal of accomplishing the most good per dollar spent does not necessarily entail giving to the charity with the highest explicitly estimated “good accomplished per dollar spent”; there may be factors that are best dealt with outside of the “explicit expected value” framework. (More at our post on the subject and the comments that followed, particularly this comment).
• We see two potential sources of value in doing explicit cost-effectiveness analysis: (a) finding relatively large and robust differences between different charitable interventions; (b) using cost-effectiveness analysis as a way to ensure that we have carefully thought through the relevant issues.

Simple cost-effectiveness measures
One way to start thinking about the relative cost-effectiveness of the three interventions is to think about the “cost per person-year of coverage,” i.e., the cost of serving one person with the intervention for one year.

• For LLINs, we estimate a cost of $1.36 per “person protected by an LLIN per year.” (This is based on$5.15 per LLIN distributed, with each LLIN covering an average of 1.8 people for an average of 2.22 years; details at the spreadsheet linked from our cost-effectiveness analysis of LLINs, column H).
• For deworming, we estimate a cost of $0.51 per person treated (details). This$0.51 figure incorporates costs that SCI, our top-rated deworming charity, reported to us. We have not fully investigated all costs associated with deworming programs (as we have for bednets and cash transfers) and believe the cost per person treated could be ~10-25% higher, leading to a cost per person treated of $.51-.64.In high-prevalence areas, deworming is annual, and all of the information we have about the impacts of deworming is based on high-prevalence areas. There is an additional question of what percentage of people treated are children, since the main case for deworming (in our view) is its developmental impacts when applied to children; we have little ability to estimate this figure and generally assume that 50% of those treated are children, leading to a figure of$1.02-$1.28 per child treated. • Cash transfers are much more difficult to model on a “per person per year” basis, since they are structured as one-time “wealth transfers.” One way of coming up with a partially informative “per person per year” figure is to look at the cost of a metal roof, which is a commonly reported use of cash transfers. GiveDirectly estimates that such a roof costs about$250 for a household and lasts about 20 years. During our visit to GiveDirectly’s operations in Kenya, recipients reported spending closer to $500 for a roof. At an average household size of 4.7 (from the “Household size analysis” document in our GiveDirectly review) and 20 years per roof, that implies about$2.66-$5.32 in recipient expenditures per “person-year of roof coverage”; at a rate of 90c transferred for each$1 in expenses, that implies about $3-$6 in donor expenses per “person-year of roof coverage.”

The obvious problem with these figures is that we don’t know how beneficial a year of LLIN coverage is relative to a year of deworming or a year of coverage by a roof. However, estimating such benefits involves a lot more judgment calls than estimating the costs. We think the figures listed above – while inconclusive and unsatisfying – are much less likely to be wildly inaccurate on their own terms than the figures that follow, so we think it’s appropriate to keep them in mind.

Humanitarian outcomes
A much more satisfying comparison, but one that is much more difficult to do with precision, is to estimate the “humanitarian value per dollar” created by each intervention. In order to do this, we need a unit of “humanitarian value” that we can standardize on. One such unit is the disability-adjusted life-year (DALY). A similar unit, which we find easier to think about but which is fundamentally similar to the DALY (and relatively easily convertible to it), is the “life saved equivalent.” For some interventions (those with direct impacts on mortality) we can estimate a “cost per life saved”; we can then try to estimate non-mortality-related benefits of interventions, decide how much we value these benefits relative to “lives saved,” and then convert them into “lives saved equivalent.”

The cost per life saved for LLIN distribution

Our current best estimate of the “cost per life saved” for LLIN distribution is about $2300 (details at the spreadsheet linked from our cost-effectiveness analysis of LLINs, column H). Some notes on this figure: • This figure is only including direct lives saved for children under five. • It is also possible that LLINs save adult lives, but figures on adult malaria deaths are disputed and we have not seen studies addressing whether LLINs save adult lives. • As discussed previously, we believe the case that LLINs have non-mortality-related “developmental benefits” is somewhat comparable to the case that deworming has such benefits. • Finally, LLIN distribution may reduce the burden of malaria, on LLIN users and on the health system, in ways not captured in the above considerations. • Our spreadsheet also estimates the cost per life saved under different assumptions about questions like “How long do LLINs last in the field?” and its overall range comes out at ~$1700-$5500. • There are several potential major sources of uncertainty in our estimate that are not captured in our spreadsheet, including the possibility of insecticide resistance, the possibility that today’s conditions differ in other ways from the conditions under which the original studies of mortality effects were done, and more. These are listed in our discussion of LLIN cost-effectiveness. The cost per “equivalent life saved” for deworming We see a lot of uncertainty around the benefits of mass deworming. We have not seen evidence that it directly saves lives, and any such lives saved are likely to be quite rare (not competitive with how often LLINs save lives). However, there is some evidence suggesting that deworming has developmental impacts: that deworming someone in childhood can cause them to earn more money later in life (among other benefits). When attempting to estimate a “cost per life saved equivalent” for deworming, one must take a view on multiple difficult-to-quantify factors, all of which are listed in the “assumptions” sheet of our spreadsheet on “cost per life saved equivalent” comparisons: • The relative value of saving a life vs. realizing the developmental benefits associated with deworming. The value assigned by the corrected Disease Control Priorities Report to averting “cognitive impairment” is 2.4% of the value it would assign to averting death, but many might consider improving a life to be more than 2.4% as valuable as saving a life. (Assumption 9 on “Assumptions” sheet) • The proportion of people dewormed by SCI who are children. (Assumption 2) • Whether the benefits of deworming are likely to scale linearly with repeated treatments. The Kenya study on developmental benefitslooks at the benefits of receiving ~2.5 years of additional treatment; it’s possible that people treated by SCI receive fewer or more years of treatment. (SCI generally aims for repeated treatment throughout childhood.) (See Assumption 3; we formalized this issue via a multiplier that captures “how helpful years of treatment in SCI’s program are, relative to the years of treatment in the key study.”) • Whether to make a “replicability adjustment” due to the fact that developmental benefits for deworming are not as robustly established as mortality effects for LLINs. (We have discussed this issue previously.) We have sought a reference point for how likely it is that a given study result will hold up under replication, and have seen some potentially relevant figures in John Ioannidis’s analysis of biomedical literature (here and here), though the analogy between biomedical studies and the studies in question has substantial limitations. We have formalized this adjustment as Assumption 4 in the sheet. • Whether to make an “external validity” adjustment, to account for the fact that infection prevalence rates were unusually high due to El Nino in the Kenya study on developmental benefits (the study that provides the most helpful information for quantifying the benefits of deworming). We have formalized this adjustment as Assumption 1 in the sheet. • How to account for the shorter-term health benefits of deworming, including both rare severe health effects (such as intestinal obstruction due to ascariasis) and subtle general health effects. Given recent developments, we think it is reasonable to consider the subtle general health benefits negligible, but we also provide the option to use our previous estimate of such benefits, which is based on the Disease Control Priorities Report. Our spreadsheet on “cost per life saved equivalent” comparisons shows the “cost per life saved equivalent” under different assumptions. We have provided many possible assumptions in the “assumptions” sheet and calculated the “cost per life saved equivalent” for deworming under each possible combination of inputs; it ranges from$31 in the most optimistic scenario to over $2 million in the most pessimistic. It also includes scenarios that represent different GiveWell staff members’ best guesses, discussed further below. Comparing LLIN distribution and deworming Recall that LLIN distribution may also have developmental benefits. Generally, the more optimistic one is about the case that health in childhood affects quality of life in adulthood, the more optimistic one ought to be about both LLIN distribution and deworming. Personally, I would guess that LLIN distribution has stronger developmental benefits than deworming on a “per-person-per-year” basis (for reasons outlined previously) though one could easily argue either side of this question. If one assumes that LLIN distribution and deworming have equal benefits on a “per-person-per-year” basis, and that the proportion of people treated who are children is similar for the two, then this implies that when considering developmental benefits alone, deworming accomplishes about 2.5x as much good per dollar as LLIN distribution (based simply on the “cost per person treated per year” from the previous section). One may also try to combine the mortality benefits and the developmental benefits of LLIN distribution under these assumptions, as we do in our spreadsheet on “cost per life saved equivalent” comparisons. Below, three of us – myself, Elie, and Alexander – explain our assumptions and give the resulting cost-effectiveness comparison. We also encourage readers to use the spreadsheet to enter their own assumptions. You can do so by going to the “Master” sheet in our spreadsheet on “cost per life saved equivalent” comparisons, manually editing columns M through V for any row, and watching the output in columns X and Y. Elie’s assumptions:$0.56 per person dewormed (assumes that our $0.51 estimate is too low by 10%, based on an underestimate we previously made of AMF’s costs before analyzing more deeply); 50% of deworming goes to children; child-years of deworming by SCI are on average 1/3 as effective as child-years in the key study; developmental benefits should be valued 20% as much as lives saved; the external validity of the key deworming study is 30.25% (based on the change in moderate-heavy infections experienced during the course of the study); 40% chance that the study would hold up under replication; standard (Disease Control Priorities Report-based) minor health benefits of deworming.$1,668 per life saved equivalent for deworming (83% from developmental benefits); $1,564 for nets. Holden’s assumptions:$0.51 per person dewormed (uses our current estimate); 50% of deworming goes to children; child-years of deworming by SCI are on average (2.41/10) as effective as child-years in the key study (this is equivalent to assuming that SCI is treating all children throughout childhood, and that benefits do not increase beyond the 2.41 years point); the external validity of the key deworming study is 30.25%; 10 people with developmental benefits are morally equivalent to 1 life saved (this figure comes from regressing my own intuitions about what’s valuable – which put very low value on saving lives relative to improving them – somewhat toward “normality,” since I’m not confident that my intuitions are appropriate on this point); 30% chance that the study would hold up under replication; standard (Disease Control Priorities Report) based minor health benefits of deworming. $3,813 per life saved equivalent for deworming (56% from developmental benefits);$2,004 per life saved equivalent for nets.

Alexander’s assumptions: $0.51 per person dewormed (uses our current estimate); 50% of deworming goes to children; child-years of deworming by SCI are on average exactly as effective as child-years in the key study (this is based on ignorance about where in the distribution of child-years of deworming additional funds might be spent, and takes into account the upside potential of later years conceivably helping to eliminate schistosomiasis from an area); the external validity of the key deworming study is 30.25%; a 2.4% quasi-disability weight for developmental effects (following the Disease Control Priorities Report, not due to confidence that it is the correct estimate but because it roughly maps to the intuition that a permanent 25% increase in income [derived from Baird et al 2012] for ~40 people starting in adulthood would be roughly as valuable as saving a life of a young person); 50% chance that study would hold up under replication; standard (Disease Control Priorities Report 3% discount rate DALY-based) minor health benefits of deworming (with no minor health benefits for bednets).$2,981 per life saved equivalent for deworming (73% from developmental benefits); $2,027 per life saved equivalent for nets. (Alexander comes up with broadly similar figures because he’s more confident in the representativeness and quality of the deworming studies than Elie or I, but places much less weight on developmental effects than either of us.) All of these figures can be converted straightforwardly to DALYs if one prefers these units. The DALYs per “life saved equivalent,” for three different versions of DALYs, are given in column AC of our spreadsheet on “cost per life saved equivalent” comparisons. Cost per “equivalent life saved” for cash? We have not attempted to use the “equivalent life saved” framework to quantify the benefits of cash transfers, since we know so little about how they are spent and what the likely results are. Instead, we compare cash transfers to the other interventions using a different framework, discussed immediately below. Financial returns The primary benefit we’ve seen evidence for, regarding deworming, is improved earnings for people dewormed in childhood. We can estimate the net present value of these benefits to arrive at a figure along the lines of “For each$1 spent on deworming, recipients receive total benefits equivalent to $X.” We can also do a modified version of this: estimating$X as a percentage of recipients’ annual income (in the case of deworming, we would guess that the percentage increase in income is a more robust figure across different settings than the dollar increase).

We can estimate a similar figure for cash transfers, using what we know about the longer-term returns to cash transfers, and compare the two. Finally, we can do a similar calculation for LLIN distribution, assuming that LLINs have similar developmental benefits to deworming (on a per-person-per-year basis), though this calculation excludes other benefits of LLINs such as direct mortality benefits.

Note: we have updated this section (and the linked spreadsheets) since initially posting to incorporate the short-term health benefits of deworming; across all scenarios we assume that each dewormed person experiences current health benefits of value equivalent to a 0.51% increase in income. This value was chosen because it results in 71% of the total benefits of deworming in our preferred scenarios (and 60% in the full set of considered scenarios) arising from developmental effects, the same as in the “lives saved equivalent” framework discussed above (assuming the short term health effects of deworming from the DCP2).

Our spreadsheet on financial returns performs this comparison. As in the previous section, many assumptions and judgment calls must be made. Regarding deworming, we have the same questions as in the previous section regarding the reliability and external validity of the studies on developmental benefits. Regarding cash transfers, there are additional questions: how much of the transfers are likely to be invested, and what rate of return are they likely to earn? What discount rate should be used to capture the fact that (a) recipients prefer present consumption to future consumption; (b) donors can invest their money and give later rather than giving today and letting recipients invest it?

Again, because this calculation is extremely sensitive to small changes in inputs, we provide the inputs for myself, Elie and Alexander. We also encourage readers to use the spreadsheet to enter their own assumptions. You can do so by going to the “Overview” sheet in our spreadsheet on financial returns, manually editing columns J through Q for any row, and watching the output in columns S through Z.

Elie’s assumptions: 50% of deworming goes to children; child-years of deworming by SCI are on average 1/3 as effective as child-years in the key study; 30.25% adjustment for external validity concerns; cash transfers are 75% invested, returning 10% a year with a 5% discount rate; 40% replicability adjustment for deworming and a 95% replicability adjustment for cash. (Elie assigns a much lower likely rate of return than seen in the studies on cash transfers, and as such does not do another large “replicability adjustment” for this.) This results in deworming being 2.3x as cost-effective as cash, with each dollar spent leading to total benefits equal to 1.25% of annual income for deworming, vs. 0.55% for cash. (In a family with the sort of income reported for GiveDirectly’s clients, this would translate to the equivalent of $2.98 in benefits for every dollar donated for deworming;$1.30 for cash.)

Holden’s assumptions: I did two different estimates, one representing low skepticism about results found in studies and another representing high skepticism. In both, I assume 50% of deworming goes to children; that child-years of deworming by SCI are on average 50% as effective as child-years in the key study; a 30.25% adjustment for external validity concerns (based on before-and-after rates of heavy-to-moderate worm infections in the key study); and a 75% investment rate (i.e., cash transfers are 75% invested vs. 25% consumed) with a 5% discount rate. However, in one case I give both the cash and deworming studies a 50% chance of holding up under replication, and use a rate of return (25%) that is triangulated from studies; in the other case I use a 5% rate of return on investment (which effectively assumes simply keeping up with the discount rate, i.e., not earning any substantive return on investment) while applying a 30% “replicability adjustment” to the deworming study. These assumptions result in deworming being ~2.9-4.2x as cost-effective as cash. In the less skeptical version, each dollar spent on deworming leads to total benefits equal to 1.93% of annual income, as opposed to 0.66% for cash; in the more skeptical version, each dollar spent on deworming leads to total benefits equal to 1.36% of annual income, as opposed to 0.32% for cash. (In a family with the sort of income reported for GiveDirectly’s clients, this would translate to the equivalent in $4.58 (optimistic)/$3.22 (skeptical) in benefits for every dollar donated for deworming; $1.57 (optimistic)/$0.76 (skeptical) for cash.)

Alexander’s assumptions are similar to those in my “optimistic” scenario except that he (as described above) assumes that 100% of the child-years of deworming are as effective as the child-years from the key study and takes the average of the reported ranges for cash transfer ROI across the two longer-term studies (54%). These go in opposite directions, leading to the conclusion that each dollar spent on deworming translates to total benefits worth 3.36% of annual income ($7.97 in a family with the sort of income reported for GiveDirectly’s clients) while each dollar spent on cash transfers translates to total benefits worth 1.21% of annual income ($2.87), which implies that deworming is ~2.8x as cost-effective as cash.

Relative to the ranges of assumptions we consider, our individual estimates come out more positive for cash transfers. The mean across the 6,075 scenarios representing each combination of our assumption set is that each dollar spent on deworming increases annual consumption by 3.37% and each dollar spent on cash transfers increases annual consumption by 0.65%, implying that deworming is ~5x as cost-effective as cash transfers.

Conclusion
Our best guess is that bednet distribution is somewhere between one and two times as cost-effective as deworming, while deworming is between 2 and 5 times as cost-effective as cash-transfers. However, we believe that there is ample room for disagreement around these figures, and we plan to write more about the way these figures should be used to guide giving decisions.

• Colin Rust on December 19, 2012 at 5:49 pm said:

I’d be interested to see you take a stab at estimating the value of the non-mortality benefits of LLIN’s. Comparing the mortality benefits alone of LLIN’s to the total benefits of other interventions is somewhat apples-to-oranges. (I see elsewhere on your site, you have a DALY estimate for LLIN’s, but it looks like it’s only based on lives saved also.)

According to WHO 2010 numbers, there are over 300 cases of malaria per malaria death (using point estimates of 655,000 malaria deaths and 216 million malaria cases in 2010; these numbers have large error bars; source: http://www.who.int/features/factfiles/malaria/en/index.html). If we assume that in 50% of the deaths averted by LLIN’s the cause of death is malaria (in many cases, malaria weakens a child who dies of some other cause that would not have been fatal otherwise), that works out to 150 cases of malaria per death averted. What would you estimate is the additional “equivalent life saved” value of avoiding those malaria cases?

As you say, another issue not covered in your estimate are the health care costs avoided by reduced malaria. Anecdotally, in extreme cases of very high malaria prevalence, funding bed nets might actually have a negative cost once you consider the cost of treating malaria.

(As I understand it, the large majority of the deaths from malaria are in children, so deaths prevented among adults are surely way within the error bars, but still it can’t hurt to get some numbers and include adult deaths averted in your best estimate.)

Of course, since AMF is already your top charity, this is arguably somewhat moot, but I think it would be good to include a more complete analysis of the benefits of your top charity.

• Stephen Weinstein on December 26, 2012 at 12:50 am said:

Increase in future earnings is a valid consideration for parents deciding how to spend their money on their own children.

For donors, it’s more complicated. If the “developmental benefit” to a child is that the child eventually grows up to get a job that otherwise would have gone to someone else, that other person loses out on that job. So the community as a whole may not receive any net economic (or other) benefit. However, this assumes that the number of good jobs is fixed, and less than the number of qualified individuals.

On the other hand, if increasing the number of children who grow up to be able to do the jobs means that a company decides to build a factory, research center, or other facility requiring a workforce, then it creates economic benefit for everyone who gets a job there, not just the treated children.

• Colin, thanks for the thoughts. I agree with you that both of the benefits you mention are benefits worth considering, and are not included in our calculation. I believe that including them in the sorts of calculations discussed in this post would be a great deal more work and would likely involve even greater uncertainty and judgment calls than the benefits we are already discussing; since AMF is our #1 charity already, we don’t feel that this analysis passes a cost-benefit test at the moment.


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### Graph

 Complex Networks Single-scale networkssimilar to a lattice in that every node has (roughly) the same degreeScale-free networksif its degree distribution, i.e., the probability that a node selected uniformly at random has a certain number of links (degree), follows a particular mathematical function called a power law.:: the fraction P(k) of nodes in the network having k connections to other nodes goes for large values of k as P(k) ~  k^{-γ} , γ is a parameter whose value is typically in the range 2 <  < 3Small-world networksby analogy with the small-world phenomenon (popularly known as six degrees of separation).with the addition of only a small number of long-range links, a regular graph, in which the diameter is proportional to the size of the network, can be transformed into a "small world" in which the average number of edges between any two vertices is very small (mathematically, it should grow as the logarithm of the size of the network), while the clustering coefficient stays large. It is known that a wide variety of abstract graphs exhibit the small-world property, e.g., random graphs and scale-free networks. Further, real world networks such as theWorld Wide Web and the metabolic network also exhibit this property.::  the typical distance L between two randomly chosen nodes (the number of steps required) grows proportionally to the logarithm of the number of nodesN in the network L ~ log NCohen and Havlin showed analytically that scale-free networks are ultra-small worlds. In this case, due to hubs, the shortest paths become significantly smaller and scale as L ~ log log NIsomorphism in Graphsan isomorphism of graphs G and H is a bijection between the vertex sets of G and Hf:V(G) -> V(H)such that any two vertices u and v of G are adjacent in G if and only if ƒ(u) and ƒ(v) are adjacent in H. This kind of bijection is commonly called "edge-preserving bijection", in accordance with the general notion of isomorphism being a structure-preserving bijection. Dual Graphhas a vertex for each plane region of G, and an edge for each edge in G joining two neighboring regions, for a certain embedding of G.symmetric: If H is a dual of G, then G is a dual of H (if G is connected)Steiner Tree Problemgiven a flow network in which each edge is given both a cost and a capacity, and a matrix of flow amounts between different pairs of terminal vertices; the task is to find a subnetwork of minimum cost whose capacities are sufficient to support a flow with the given flow amounts between any pair of terminals. When the flow amounts are all equal, this reduces to the classical Steiner tree problem.Laplacian Matrixis the difference of the degree matrix D and the adjacency matrix A of the graph. In the case of directed graphs, either the indegree or outdegree might be used, depending on the application.  So it's value is the degree of the node on the diagonal and -1 if those two nodes are adjacent. [2 0 0 0 0 0]           [0 1 0 0 1 0]|0 3 0 0 0 0|            |1 0 1 0 1 0||0 0 2 0 0 0|     -      |0 1 0 1 0 0||0 0 0 3 0 0|            |0 0 1 0 1 1||0 0 0 0 3 0|            |1 1 0 1 0 0|[0 0 0 0 0 1]           [0 0 0 1 0 0]For a graph G and its Laplacian matrix L with eigenvalues  λ0 <= λ1 <= ...<= λn-1, L is always positive-semidefinite (∀i, λi >= 0, λ0 = 0)The number of times 0 appears as an eigenvalue in the Laplacian is the number of connected components in the graph.λ0 = 0 (because every Laplacian matrix has an eigenvector v0 = [1 1 ... 1] that, for each row, adds the corresponding node's degree (from the diagonal) to a "-1" for each neighbor so that L v0 = 0)The second smallest eigenvalue of L is the algebraic connectivity (or Fiedler value) of G.Algebraic ConnectivityThe second smallest eigenvalue of the laplacian matrix of a graph. It is greater than zero if and only if the graph is connected.This is a corollary to the fact that the number of times 0 appears as an eigenvalue in the Laplacian is the number of connected components in the graph. The magnitude of this value reflects how well connected the overall graph is, and has been used in analysing the robustness and synchronizability of networks.ConnectivityIt asks for the minimum number of elements (nodes or edges) which need to be removed to disconnect the remaining nodes from each other.• Weakly connected: if changing directed edges of a graph to undirected makes it connected.• Connected: if there is a directed path from u to v for every pair• Strongly connected: if there is a directed path from u to v and from v to u for every pair• k-connectivity: if there does not exist a set of k-1 vertices whose removal disconnects the graph.• Edge connectivity: specific edge would disconnect the graph, that edge is called a bridge. More generally, the edge cut of G is a group of edges whose total removal renders the graph disconnected. • The greatest number of independent paths between u and v is written as κ′(u,v), and the greatest number of edge-independent paths between u and v is written as λ′(u,v).It is closely related to the theory of network flow problems. It is an important measure of its robustness as a network.Network Flowfor each edge of the graph there is a capacity that it can handle and there is a flow that can flow through it. flow <= capacity. flow-in=flow-out unless source or destination. flow-in= - flow-out
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# 1. All categories together constitute the taxonomic category. 2. Each category represents a rank and is commonly termed as taxon

( a ) 1 is incorrect but 2 is correct
( b ) Both are correct
( c ) Both are incorrect
( d ) 1 is correct but 2 is incorrect
Can you answer this question?

## 1 Answer

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(A) is the right answer.

The category is a part of overall taxonomic arrangement, it is called the taxonomic category and all categories together constitute the taxonomic hierarchy. Each category, referred to as a unit of classification, in fact, represents a rank and is commonly termed as taxon (pl.: taxa).
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# Math Help - Permutation/Combination Problem Help

1. ## Permutation/Combination Problem Help

Hello, all.

I got the problem

(a) xP2

(b)
xC2

The answers in the back of the book say (a)
X2-X and (b) X2-X over 2

I can do these problems normally, but when they threw the X in there, it confused me greatly. Anyone able to lead me in the right direction?

--Thanks

2. Originally Posted by HardyHar
Hello, all.

I got the problem

(a) xP2

(b)
xC2

The answers in the back of the book say (a)
X2-X and (b) X2-X over 2

I can do these problems normally, but when they threw the X in there, it confused me greatly. Anyone able to lead me in the right direction?

--Thanks
The x shouldn't be much of an issue...

$_x P_2=\frac{x!}{(x-2)!}=\dots$

$_x C_2=\frac{x!}{2!(x-2)!}=\dots$

Just note that $x!=x(x-1)(x-2)!$

Can you take it from here??

--Chris

3. Originally Posted by Chris L T521
The x shouldn't be much of an issue...

$_x P_2=\frac{x!}{(x-2)!}=\dots$

$_x C_2=\frac{x!}{2!(x-2)!}=\dots$

Just note that $x!=x(x-1)(x-2)!$

Can you take it from here??

--Chris
Yes, sir. Thank you.


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# How To Improve Your Luck

In September 1928 a bacteriologist name Alexander Fleming came back from a "holiday" (vacation) to find a set of petri dishes which had been "necessarily exposed to the air" had become contaminated by various micro-organisms. One of those petri dishes had grown a mold that appeared to be inhibiting the growth of a colony of Staphylococcus aureus. (You can read the original paper here)

To quote Dr. Fleming: “When I woke up just after dawn on September 28, 1928, I certainly didn’t plan to revolutionize all medicine by discovering the world’s first antibiotic, or bacteria killer."

You've probably heard this story before. However, there is a second part to the penicillin story that is told a lot less.

What happened following the discovery on the part of Dr. Fleming was practically nothing. He published a paper on his findings, and that was pretty much it. Sure he did some more research in the area, but he ultimately concluded that his discovery wouldn't be useful for treating infection and wasn't worth pursuing further. Effectivity his part in the story stopped right after publication.

I want to pause to let that sink in for a second.

Not many people realize that it is entirely possible that the publication could have been the end of the story... or worse it might not have even made publication. If that were the case no one would would have benefited from perhaps the greatest life saving discovery in the 20th century.

What did happen (a decade after publication) was that an Oxford Scientist named Dr. Howard Florey was reading back issues of The Journal of Experimental Pathology and ran into Fleming's article. It interested him enough to try and reproduce the work, eventually culminating in a streptococcus study of mice in 1940 with Dr. Ernst Chain & Dr. Norman Heatley. That study produced verifiable results that the production of antibiotics were indeed possible (though engineering efforts were required to make mass production viable).

### How narratives shape our perception

Dr. Fleming's story is clearly in the genre of "dumb luck" (supposedly he even used the most famous phrase in all of science: "Huh. That's funny..."). Sure, he was a trained bacteriologist, and he had been working on infectious disease studies for a long time, but the mold literally fell out of the air perfectly into his petri dish.

Dr. Florey's is on the other side of the "narrative" coin. He built a top lab at Oxford through hustle and grit. According to a PBS article on Florey he was "a master at extracting research grants from tight-fisted bureaucrats and an absolute wizard at administering a large laboratory filled with talented but quirky scientists." His "discovery" of the Fleming study was the result of searching, learning, and accumulating resources to be able to leverage.

These two versions of the "luck narrative" form a dichotomy for the way people like to look at luck. Either 1) serendipity is out of our control and "the gods will gift who they deem worthy" or 2) we can make our own luck through hard work, and determination.

### What is luck anyway?

So the following is what is called the "Probability Mass Function" (one way of defining "luck"):

$$f(x,n,p) = \frac{!x}{n!(x-n)!}p^n(1-p)^x-n$$

# Defined
p = probablility of success
x = is the number of attempts
n = the number of winning attempts / successful attempts

To be "lucky" in the mathematical sense is where an agent achieves a successful (or unsuccessful) outcome that exceeds its probabilistic expectation given an environment. (link)

"Luck", in the colloquial sense, is the result of chance. It's a surprise, an unexpected event. You have "bad" luck when bad (unexpected) things happen to you or "good" luck when good things happen to you. Luck in this sense is all around us, impacts us all, with the only difference being how much it effects us.

[This colloquial luck has powerful psychological effects too... Higher socioeconomic status is correlated with lower belief in luck and vice-versa (e.g. research). ]

## How can one improve their luck?

So how can you improve your luck?

...Well, you can't. Thanks for reading my article! (just kidding)

That is to say, you can't improve your luck in a "mathematical sense." Luck in that sense is actually random... otherwise it wouldn't be luck.

"Luck" in the colloquial sense however is arguably more about narrative than probability. The magnitude of the outcomes in "colloquial luck" create the perceived "luckiness." Receiving an "asymmetric payoff" makes you appear to be more lucky. So if you are interested in improving your luck in the colloquial sense you can accomplish this by exposing yourself to environments that have outsized payouts.

...Ergo the process of "Improving your luck" is simple. Marry together what we know about mathematical luck and colloquial luck.

Start by finding environments that can provide outsized payouts. I cannot stress enough how important it is to FIND THE RIGHT ENVIRONMENT, as it will basically be the single largest influence on your probability of success. Searching for these sorts of environments takes work. You need to figure out as accurately as possible what your probability of success will be, so that you can commit to working in that environment... which is no easy task.

After you have chosen the right environment you want to search for ways to improve the probability of being successful and execute them. More often than not the simplest way to improve the probability of a successful outcome is just to increase the number of attempts. But similarly, there are often ways to decrease the total number of attempts needed to be successful- basically influence your environment in a way that is favorable to you.

And of course the last piece of the puzzle is to figure out ways to increase the narrative value when you are successful. Being perceived as being lucky is a powerful attribute and can help you improve your "luck" for future opportunities.

### Hundreds of millions of lives saved...

When looking back at Dr. Howard Florey, his search through back issues of the Journal of Experimental Pathology was effectively his way of choosing the environment he wanted to work in. He was beginning in a place that already had some level of success and he knew he could increase the chances of success by starting with something that was already somewhat successful. The work of Alexander Fleming wasn't just a starting place- it reduced the total number of attempts it would take to be successful... all Florey had to do was put the work in and there was a good chance that luck would find him. The final piece of the puzzle was that Florey was choosing to work on something that had an outsized payout or impact.

...If you want it distilled, here's your recipe for improving your luck:

1. Select what you work on carefully.
2. Make sure that you can be successful and that the payouts are outsized.
3. Try to find ways to reduce what it takes to be successful.
4. Be focused and work hard.

If you liked the article you should check out my monthly newsletter "Cocktail Napkin Math":


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Setting up a Single Point Calculation

As mentioned on the welcome page, FHI-aims and AiiDA interact through the aiida-ase plugin. This has the advantage that existing Python scripts, which already exploit the benefits of using FHI-aims with ASE, can easily be extended for interoperability with AiiDA.

This tutorials assumes a basic understanding of how to interact with AiiDA. Make sure to check out the official AiiDA documentation which features many examples on setting up first calculations in AiiDA, tracking their provenance, and inspecting the results.

That being said, let us take a look at a simple python script which performs a single point calculation for silicon under periodic boundary conditions. All AiiDA calculations follow a similar scheme:

1. Load the relevant AiiDA and ASE modules: As is customary in Python, all required modules are imported at the beginning of the script.

from aiida.orm import load_code
from aiida.plugins import DataFactory, CalculationFactory
from aiida.engine import submit

# We will create the structure from scratch using ASE's build features.
from ase.build import bulk

2. Instantiate AiiDA classes: Aiida provides convenient classes to store different types of data in its repository or database infrastructure. For example, the Dict data class wraps Python dictionaries and is typically used to store calculation parameters. Moreover, AiiDA comes with specialized data types to store chemical structure information (structure) and kpoints (array.kpoints). The DataFactory method is used here to instantiate the classes.

Dict = DataFactory('dict')
StructureData = DataFactory('structure')
KpointsData = DataFactory('array.kpoints')

Similarily, the CalculationFactory method makes it easy to load a calculation plugin:

AseCalculation = CalculationFactory('ase.ase')

3. Define the structure: In this case, we are using the bulk module provided by ASE to generate a simple bulk silicon unit cell with periodic boundary conditions.

atoms = bulk('Si')

4. Set calculation parameters. Just like the ASE interface to FHI-aims, aiida-ase understands all FHI-aims keywords which are documented in the manual. In AiiDA, these parameters are usually summarized in a nested dictionary. Here, we are loading the aims.Aims calculator (which is how the FHI-aims calculator module in ASE is internally called) and providing it with several arguments. Most importantly, one has to specify the name of the compiled FHI-aims executable (as it stands in the \$PATH), the desired name of the output file, the path to the species directory, and the exchange-correlation functional. Furthermore, we are here setting custom values for the convergence criteria.

parameters = {
'calculator': {
'name': 'aims.Aims',
'args': {
'aims_command': 'AIMS_EXECUTABLE',
'outfilename': 'aims.out',
'species_dir': 'PATH/TO/SPECIES/DIRECTORY',
'xc': 'PBE',
'sc_accuracy_etot': 1e-4,
'sc_accuracy_eev': 1e-2,
'sc_accuracy_rho': 1e-5,
'sc_accuracy_forces': 1e-4
}
}
}

Apart from that, we are declaring an instance of the AiiDA KPointsData class and constructing a 2x2x2 uniformily spaced kpoint grid in the Brillouin zone.

kpoints = KpointsData()
kpoints.set_kpoints_mesh([2, 2, 2])

5. Set calculation metadata: Metadata in AiiDA has two major jobs: First, it allows the user to give a label and description to every calculation, ensuring it may easily be retrieved from the database in the future. Second, metadata is also used to inform the scheduling system on the worker machine about the desired resources for this job. In our case, we are requesting a single MPI process without distribution, and setting a (way too large) wallclock time.

metadata = {
'label': 'Si with FHIAims',
'description': 'Test calculation of FHI-aims using aiida-ase.',
'options': {
'resources': {
'num_machines': 1,
'num_mpiprocs_per_machine': 1
},
'max_wallclock_seconds': 30 * 60,  # 30 minutes
}
}

6. Putting the pieces together: All calculation details have to be combined into one calculation object. This can either be achieved by constructing a large dictionary containing all the information, or by using the AiiDA builder object which will create such a dictionary automatically.

codes = load_code('ASECODELABEL@COMPUTER')
builder = codes.get_builder()
builder.structure = StructureData(ase=atoms)
builder.kpoints = kpoints
builder.parameters = Dict(dict=parameters)

7. Submit the calculation to the AiiDA queue.

submit(builder)

After the calculation has been submitted, the AiiDA daemon will take care of sending it to a possible remote server, checking the calculation status, retrieving the desired results and handling errors, if any occurred.

Below you can find a complete version of the submission script.

"""FHI-aims-driven SPC on bulk silicon using AiiDA."""

from aiida.engine import submit
from aiida.plugins import DataFactory, CalculationFactory

# ASE imports for structure building.
from ase.build import bulk

Dict = DataFactory('dict')
StructureData = DataFactory('structure')
KpointsData = DataFactory('array.kpoints')
AseCalculation = CalculationFactory('ase.ase')

# Change the following value to the ASE code configured under the name
# ASECODELABEL on machine COMPUTERLABEL.
code = 'ASECODELABEL@COMPUTERLABEL'

# Set calculation details.
parameters = {
'calculator': {
'name': 'aims.Aims',
'args': {
'aims_command': 'AIMS_EXECUTABLE',
'outfilename': 'aims.out',
'species_dir': 'PATH/TO/SPECIES/DIRECTORY',
'xc': 'PBE',
'sc_accuracy_etot': 1e-4,
'sc_accuracy_eev': 1e-2,
'sc_accuracy_rho': 1e-4,
'sc_accuracy_forces': 1e-3
}
}
}

'label': 'Si with FHIAims',
'description': 'Test calculation of FHI-aims using aiida-ase.',
'options': {
'resources': {
'num_machines': 1,
'num_mpiprocs_per_machine': 1
},
'max_wallclock_seconds': 30 * 60,  # 30 minutes
}
}

# Arrange 2x2x2 k-point grid.
kpoints = KpointsData()
kpoints.set_kpoints_mesh([2, 2, 2])

# Use the builder object to string together all calculation details.