Title | Calculating power by simulation | |

Author | Alan Feiveson, NASA |

First, the general procedure is

- Use the underlying model to generate random data with (a) specified sample sizes, (b) parameter values that express the difference one is trying to detect with the hypothesis test, and (c) nuisance parameters such as variances.
- Run the Stata estimation program (
**regress**,**glm**, etc.) on these randomly generated data. - Get the parameter estimates and standard errors using
**e(b)**and**e(V)**. - Calculate the test statistic and
*p*-value. For example, if it is a*z*-test, divide the parameter estimate of interest by its standard error and use the**normal()**function to get the*p*-value. Specifically, suppose b is the coefficient estimate with standard error s_{b}. The two-sided*p*-value to test the hypothesis of a zero coefficient is then equal to 2[1−phi( | t | ) ], where t = b/s_{b}and phi is the standard normal distribution function returned by**normal()**. - Do Steps 1–4 many times, say,
*N*, and save the*p*-values in a file with*N*observations. - The estimated power for a level alpha test is simply the proportion of
observations (out of
*N*) for which the*p*-value is less than alpha.

**Warning:** If there are
covariates, make sure they remain fixed throughout all *N* iterations
of the simulation! Do not regenerate them each time.

Note: As a check, always run the simulation for the null case to make sure
the power comes out to be alpha (to within the sampling error of the number
of iterations). More generally, in the null case, the distribution of the
*p*-values should be uniform. This may be easily tested in Stata using
the **ksmirnov** command.

Now an example: We wish to test the effect of a drug dose on a
Poisson-distributed response *y* in rats. For a sample of *n*
rats, we have the Poisson regression model

logmu=_{i}b+_{0}b(_{i}x_{i}i= 1,2,...,n)y~ Poisson(mu)

where *x _{i}* is the dose in milligrams given to the

/* poispow.do - does power estimation for Poisson regression model version 14.1 model: log(mu) = b0 + b1*x y ~ Poisson(mu) Specifically, power is estimated for testing the hypothesis b1 = 0, against a user-specified alternative for a user-specified sample size. Without loss of generality, we can assume the true value of b0 is 0. In the `args' statement below: N is number of iterations in the simulation r is the number of rats receiving the jth dose (j=1,3) d1, d2 and d3 are the fixed dosages b1 is the "true" value of b1 (the alternative hypothesis). */ args N r d1 d2 d3 b1 drop _all set obs 3 generate x=`d1' in 1 replace x=`d2' in 2 replace x=`d3' in 3 expand `r' sort x generate mu=exp(0 + `b1'*x) /* (the "zero" is to show b0 = 0) */ save tempx, replace /* Note: Here, I generated my "x" and mu-values and stored them in a dataset -tempx- so that the same values could be used throughout the simulation */ /* set up a dataset with N observations which will eventually contain N "p"-values obtained by testing b1=0. */ drop _all set obs `N' generate pv = . /* Loop through the simulations */ local i 0 while `i' < `N' { local i=`i'+1 preserve use tempx,clear /* get the n = 3*r observations of x and the mean mu into memory */ gen xp = rpoisson(mu) /* generate n obs. of a Poisson(mu)random variable in variable -xp- */ quietly poisson xp x /* do the Poisson regression */ matrix V=e(V) /* get the standard-error matrix */ matrix b=e(b) /* get the vector of estimated coefficients */ scalar tv=b[1,1]/sqrt(V[1,1]) /* the "t"-ratio */ scalar pv = 2*(1-normal(abs(tv))) /* the p-value */ restore /* get back original dataset with N observations */ quietly replace pv=scalar(pv) in `i' /* set pv to the p-value for the ith simulation */ _dots `i' 0 } /*The dataset in memory now contains N simulated p-values. To get an estimate of the power, say for alpha=.05, just calculate the proportion of pv's that are less than 0.05: */ count if pv<.05 scalar power=r(N)/`N' scalar n=3*`r' noisily display "Power is " %8.4f scalar(power) " for a sample size of " /* */ scalar(n) " and alpha = .05" exit

We now run this program for various values of *r* (10, 20, 30, 40, 50,
60, and 70), with 100 simulations per run and with *α* = 0.05
hard coded in (of course this could easily be changed or made a user input).

. set seed 1234 . run poispow 100 10 .2 .5 1.0 0.64Power is 0.3100 for a sample size of 30 and alpha = .05. run poispow 100 20 .2 .5 1.0 0.64Power is 0.5700 for a sample size of 60 and alpha = .05. run poispow 100 30 .2 .5 1.0 0.64Power is 0.7300 for a sample size of 90 and alpha = .05. run poispow 100 40 .2 .5 1.0 0.64Power is 0.8700 for a sample size of 120 and alpha = .05. run poispow 100 50 .2 .5 1.0 0.64Power is 0.9200 for a sample size of 150 and alpha = .05. run poispow 100 60 .2 .5 1.0 0.64Power is 0.9800 for a sample size of 180 and alpha = .05. run poispow 100 70 .2 .5 1.0 0.64Power is 0.9700 for a sample size of 210 and alpha = .05

To summarize, we have the following:

r n power (est.) 10 30 0.3100 20 60 0.5700 30 90 0.7300 40 120 0.8700 50 150 0.9200 60 180 0.9800 70 210 0.9700

Although each estimate is not that precise, being based on only 100 trials,
one could fit a logistic or other regression curve through these data to get
a good estimate of which value of *r* would give a prescribed power,
such as 0.80. Of course, one could get better accuracy by increasing
*N* at the expense of more computer time.

Finally (perhaps this should have been done first), we run the program with
*b _{1}* = 0 and a large value of

. set seed 1234 . run poispow 1000 10 .2 .5 1.0 0.0Power is 0.0430 for a sample size of 30 and alpha = .05

There were 43 simulated *p*-values less than 0.05, which for 1000
binomial trials gives a 95% confidence interval of about (0.0313, 0.0575) for
the level of the test (see **cii** command results below). This suggests
that the normal approximation *z*-test is quite good for 30
observations (*r* = 10).

. cii proportions 1000 43

-- Binomial Exact -- | ||

Variable | Obs Proportion Std. Err. [95% Conf. Interval] | |

1,000 .043 .0064149 .0312912 .0574863 |

**Note:** Users of Stata 14.0 and earlier should use the syntax **cii 1000 43**
instead of **cii proportions 1000 43**.

Finally, we check the entire distribution of simulated *p*-values for a
uniform distribution:

. ksmirnov pv=pvOne-sample Kolmogorov-Smirnov test against theoretical distribution pv

Smaller group D P-value |

pv: 0.0055 0.941 |

Cumulative: -0.0372 0.063 |

Combined K-S: 0.0372 0.126 |

It passes with flying colors. Here is the histogram of *p*-values with
32 bins:

. histogram pv, bin(32)(bin=32, start=.00449199, width=.03110963)