# Power for Small Sample Exact Test A helper function to calculate power for different alternative distributions ```r powalt(SST, p_alt, a = 0.05) ``` ## Arguments - `SST`: a small_samptest object created with the small_samptest function - `p_alt`: vector of alternative probabilities to calculate power for - `a`: scaler, alpha level for power estimate, default 0.05 ## Returns A PowerSmallSamp object with slots for: * `permutations`, a dataframe that contains the exact probabilities and test statistic values for every possible permutation * `power`, the estimated power of the scenario * `alternative`, the alternative distribution of probabilities specified * `null`, the null distribution (taken from the SST object) * `alpha`, the specified alpha level ## Details This construct a null distribution for small sample statistics for N counts in M bins. Example use cases are to see if a repeat offender have a proclivity to commit crimes on a particular day of the week (see the referenced paper). It can also be used for Benford's analysis of leading/trailing digits for small samples. ## Examples ```r # Counts for different days of the week d <- c(3,1,2,0,0,0,0) #format N observations in M bins res <- small_samptest(d=d,type="G") # Power if someone only commits crime on 4 days of the week alt_p <- c(1/4,1/4,1/4,1/4,0,0,0) rp <- powalt(res,alt_p) #need to use previously created SST object print(rp) # Example for Benfords analysis f <- 1:9 p_fd <- log10(1 + (1/f)) #first digit probabilities #check data from Nigrini page 84 checks <- c(1927.48,27902.31,86241.90,72117.46,81321.75,97473.96, 93249.11,89658.17,87776.89,92105.83,79949.16,87602.93, 96879.27,91806.47,84991.67,90831.83,93766.67,88338.72, 94639.49,83709.28,96412.21,88432.86,71552.16) # To make example run a bit faster checks <- checks[1:10] # extracting the first digits fd <- substr(format(checks,trim=TRUE),1,1) tot <- table(factor(fd, levels=paste(f))) resG <- small_samptest(d=tot,p=p_fd,type="Chi") # Lets look at alt under equal probabilities (very conservative) alt_equal <- rep(1/length(p_fd),length(p_fd)) powalt(resG,alt_equal) ``` ## See Also `small_samptest()` for generating the SST object needed to estimate the power

powalt function