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fuller and mara drita 2001 a retrospective view of corporate diversity training from 1964 to the present anand and winters 2008 best practices or best guesses assessing the efficacy of corporate affirmative action and diversity policies kalev dobbin and kelly 2006 mixed signals the unintended effects of diversity initiatives dover et al 2020 have we moved beyond the civil rights revolution skrentny 2014 legal environments and organizational governance the expansion of due process in the american workplace edelman 1990 legal ambiguity and the politics of compliance affirmative action officers dilemma edelman et al 1991 the social construction of race race categorization and the regulation of business and science lee and skrentny 2010 making hispanics mora 2014 inventing race skrentny 2002 firm responses to activism the nixon in china effect activism imitation and the institutionalization of contentious practices briscoe and safford 2008 a political mediation model of corporate response to social movement activism king 2008 the politics of alignment and the quiet transgender revolution in fortune 500 corporations 2008 to 2017 ghosh 2021 corporate social responsibility and social control social responsibility messages and worker wage requirements field experimental evidence from online labor marketplaces burbano 2016 corporate social responsibility as an employee governance tool evidence from a quasi experiment flammer and luo 2017 k 12 education contentious curricula binder 2009 july 14 2022 at 6 30 pm gr resampling approach to power analysis a coauthor and i are doing power analyses based on a pilot test and we quickly realized that it s really hard to calculate a power analysis for anything much more exotic than a t test of means as i usually do when there s no obvious solution i decided to just brute force it with a monte carlo approach the algorithm is as follows start with pilot data draw a sample with replacement of the target sample size n and do this trials times run the estimation with each of these resampled datasets and see if enough of them achieve significance with conventional power and alpha this means 80 have p 05 in theory this approach should allow a power analysis for any estimator no matter how strange i was really excited about this and figured we could get a methods piece out of it until my coauthor pointed out green and macleod beat us to it nonetheless i figured it s still worth a blogpost in case you want to do something that doesn t fit within the lme4 package around which green and macleod s simr package wraps the above algorithm for instance as seen below i show how you can do a power analysis for a stratified sample import pre test data in a real workflow you would just use some minor variation on pilot read_csv pilot csv but as an illustration we can assume a pilot study with a binary treatment a binary dv and n 6 let us assume that 1 3 of the control group are positive for the outcome but 2 3 of the treatment group note that a real pilot should be bigger i certainly wouldn t trust a pilot with n 6 treatment c 0 0 0 1 1 1 outcome c 0 0 1 0 1 1 pilot as data frame cbind treatment outcome i suggest selecting just the variables you need to save memory given that you ll be making many copies of the data in this case it s superfluous as there are no other variables but i m including it here to illustrate the workflow pilot_small pilot select treatment outcome display observed results as a first step do the analysis on the pilot data itself this gives you a good baseline and also helps you see what the estimation object looks like unfortunately this varies considerably for different estimation functions est_emp glm outcome treatment data pilot family binomial summary z_emp est_emp coefficients 3 est_emp call glm formula outcome treatment family binomial data pilot deviance residuals 1 2 3 4 5 6 0 9005 0 9005 1 4823 1 4823 0 9005 0 9005 coefficients estimate std error z value pr z intercept 0 6931 1 2247 0 566 0 571 treatment 1 3863 1 7321 0 800 0 423 dispersion parameter for binomial family taken to be 1 null deviance 8 3178 on 5 degrees of freedom residual deviance 7 6382 on 4 degrees of freedom aic 11 638 number of fisher scoring iterations 4 note that we are most interested in the t or z column in the case of glm this is the third column of the coefficient object but for other estimators it may be a different column or you may have to create it manually by dividing the estimates column by the standard error column now you need need to see how the z column appears when conceptualized as a vector look at the values and see the corresponding places in the table i wrapped it in as vector because some estimators give z as a matrix as vector z_emp 1 0 5659524 0 8003776 in this case we can see that z_emp 1 is z for the intercept and z_emp 2 is z for the treatment effect obviously the latter is more interesting set range of assumptions for resamples now we need to set a range of assumptions for the resampling trials is the number of times you want to test each sample size higher values for trials are slower but make the results more reliable i suggest starting with 100 or 1000 for exploratory purposes and then going to 10 000 once you re pretty sure you have a good value and want to confirm it the arithmetic is much simpler if you stick with powers of ten nrange is a vector of values you want to test out note that the z value for your pilot gives you a hint if it s about 2 you should try values similar to those in the pilot if it s much smaller than 2 you should try values much bigger trials 1000 how many resamples per sample size nrange c 50 60 70 80 90 100 values of sample size to test set up resampled data do the regressions and store the results this is the main part of the script it creates the resampled datasets in a list called dflist the list is initialized empty and dataframes are stored in the list as they are generated store z scores and sample size in matrix results dflist list k 1 this object keeps track of which row of the results object to write to results matrix nrow length nrange trials ncol 3 data frame adjust ncol value to be length as vector z_emp 1 colnames results c int treatment n replace the vector with names for as vector z_emp positions followed by n the nanes need not match the names in the regression table but should capture the same concepts for i in 1 length nrange dflist i list for j in 1 trials dflist i j sample_n pilot_small size nrange i replace t est glm outcome treatment data dflist i j family binomial summary adjust the estimation to be similar to whatever you did in the test estimation block of code just using data dflist i j instead of data pilot z est coefficients 3 you may need to tweak this line if not using glm results k c as vector z nrange i k k 1 create vector summarizing each resample as significant 1 or not significant 0 results treatment stars 0 results treatment stars abs results treatment 1 96 1 interpret results dist of z by sample size as an optional first step plot the distributions of z scores across resamples by sample size results ggplot mapping aes x treatment geom_density alpha 0 4 theme_classic facet_wrap n number of significant resamples for treatment effect next make a table for what you really want to know which is how often resamples of a given sample size gives you statistical significance this rate can be interpreted as power table results n results treatment stars 0 1 50 355 645 60 252 748 70 195 805 80 137 863 90 105 895 100 66 934 as you can see n 70 seems to give about 80 power to confirm this and get a more precise value you d probably want to run the script again but this time with nrange c 67 68 69 70 71 72 73 and trials 10000 stratified samples 50 50 strata common for rfts note that you can modify the approach slightly to have stratified resamples for instance you might want to ensure an equal number of treatment and outcome cases in each resample to mirror a 50 50 random assignment design this should mostly be an issue for relatively small resamples as for large resamples you are likely to get very close to the ratio in the pilot test just by chance to do this we modify the algorithm by first splitting the pilot data into treatment and control data frames and then sampling separately from each before recombining but otherwise using the same approach as before pilot_control pilot_small filter treatment 0 pilot_treatment pilot_small filter treatment 1 trials_5050 1000 how many resamples per sample size nrange_5050 c 50 60 70 80 90 100 values of sample size to test nrange_5050 2 round nrange_5050 2 ensure all values of nrange are even dflist_5050 list k 1 this object keeps track of which row of the results object to write to results_5050 matrix nrow length nrange_5050 trials_5050 ncol 3 data frame adjust ncol value to be length as vector z_emp 1 colnames results_5050 c int treatment n replace the vector with names for as vector z_emp positions followed by n the nanes need not match the names in the regression table but should capture the same concepts for i in 1 length nrange_5050 dflist_5050 i list for j in 1 trials_5050 dflist_5050 i j rbind sample_n pilot_control size nrange_5050 i 2 replace t sample_n pilot_treatment size nrange_5050 i 2 replace t est glm outcome treatment data dflist_5050 i j family binomial summary z est coefficients 3 results_5050 k c as vector z nrange_5050 i k k 1 results_5050 treatment stars 0 results_5050 treatment stars abs results_5050 treatment 1 96 1 number of significant resamples for treatment effect with 50 50 strata table results_5050 n results_5050 treatment stars 0 1 50 330 670 60 248 752 70 201 799 80 141 859 90 91 909 100 63 937 not surprisingly our 80 power estimate is still about 70 one fixed strata and the other estimated or perhaps you know the size of a sample in one strata and want to test the necessary size of another strata perhaps a power analysis for a hypothesis specific to strata one gives n 1 as its necessary sample size but you want to estimate a power analysis for a pooled sample where n n 1 n 2 likewise you may wish to estimate the necessary size of an oversample note that if you already have one strata in hand you could modify this code to work but should just use the data for that strata not resamples of it for one fixed and one estimated strata let s assume our pilot test is departments a and b from the ucbadmissions file that we know we need n 500 for 80 power on some hypothesis specific to department a and we are trying to determine how many we need for department b in order to pool and analyze them together i specify dummies for gender male and a dummy for department a vs b ucb_tidy ucbadmissions as_tibble uncount mutate male gender male admitted admit admitted select male admitted dept ucb_a ucb_tidy filter dept a mutate depa 1 ucb_b ucb_tidy filter dept b mutate depa 0 n_a 500 trials_b 1000 how many resamples per sample size nrange_b c 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 values of sample size to test dflist_b list k 1 this object keeps track of which row of the results object to write to results_b matrix nrow length nrange_b trials_b ncol 4 data frame adjust ncol value to be length as vector z_emp 1 colnames results_b c int male depa n replace the vector with names for as vector z_emp positions followed by n the nanes need not match the names in the regression table but should capture the same concepts for i in 1 length nrange_b dflist_b i list for j in 1 trials_b dflist_b i j rbind sample_n ucb_b size nrange_b i replace t sample_n ucb_a size n_a replace t est glm admitted male depa data dflist_b i j family binomial summary z est coefficients 3 results_b k c as vector z nrange_b i k k 1 add dummy for dept results_b male stars 0 results_b male stars abs results_b male 1 96 1 number of significant resamples for gender effect with unbalanced strata table results_b n results_b male stars 0 1 50 108 892 100 132 868 150 128 872 200 120 880 250 144 856 300 133 867 350 153 847 400 194 806 450 181 819 500 159 841 550 186 814 600 168 832 650 185 815 700 197 803 750 201 799 800 194 806 850 191 809 900 209 791 950 183 817 1000 214 786 this reveals a tricky pattern we see about 90 power when there are either 50 or 100 cases from department b i e 550 600 total including the 500 from department a with trials_b 1000 it s a bit noisy but still apparent that the power drops as we add cases from b and then rises again along a u shaped curve normally you d expect that more sample size would mean more statistical power because standard error is inversely proportional to the square root of degrees of freedom the trick is that this assumes nothing happens to β as it happens ucbadmissions is a famous example of simpson s paradox and specifically the gender effects are much stronger for department a glm admitted male data ucb_a family binomial summary call glm formula admitted male family binomial data ucb_a deviance residuals min 1q median 3q max 1 8642 1 3922 0 9768 0 9768 0 9768 coefficients estimate std error z value pr z intercept 1 5442 0 2527 6 110 9 94e 10 maletrue 1 0521 0 2627 4 005 6 21e 05 signif codes 0 0 001 0 01 0 05 0 1 1 dispersion parameter for binomial family taken to be 1 null deviance 1214 7 on 932 degrees of freedom residual deviance 1195 7 on 931 degrees of freedom aic 1199 7 number of fisher scoring iterations 4 than the gender effects are for department b glm admitted male data ucb_b family binomial summary call glm formula admitted male family binomial data ucb_b deviance residuals min 1q median 3q max 1 5096 1 4108 0 9607 0 9607 0 9607 coefficients estimate std error z value pr z intercept 0 7538 0 4287 1 758 0 0787 maletrue 0 2200 0 4376 0 503 0 6151 signif codes 0 0 001 0 01 0 05 0 1 1 dispersion parameter for binomial family taken to be 1 null deviance 769 42 on 584 degrees of freedom residual deviance 769 16 on 583 degrees of freedom aic 773 16 number of fisher scoring iterations 4 specifically department a strongly prefers to admit women whereas department b has only a weak preference for admitting women the pooled model has a dummy to account for department a generally being much less selective than department b but it tacitly assumes that the gender effect is the same as it has no interaction effect this means that as we increase the size of the department b resample we re effectively flattening the gender slope through compositional shifts towards department b and its weaker preference for women december 3 2021 at 2 14 pm gr dungeon crawling together i ve been reading a lot of osr games and they remind me a lot of the traditionalist phase of the genre trajectory model from lena and peterson s 2008 asr and lena s book banding together osr games are attempts to recreate dungeons and dragons as it was played in the 1970s often by using the ogl think creative commons or gpl for the 2000s version of the game but ...
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