getDesignLogistic {lrstat}

Power and sample size for logistic regression

Description

Obtains the power given sample size or obtains the sample size given power for logistic regression of a binary response given the covariate of interest and other covariates.

Usage

getDesignLogistic(
  beta = NA_real_,
  n = NA_real_,
  ncovariates = NA_integer_,
  nconfigs = NA_integer_,
  x = NA_real_,
  pconfigs = NA_real_,
  corr = 0,
  oddsratios = NA_real_,
  responseprob = NA_real_,
  rounding = TRUE,
  alpha = 0.05
)

Arguments

beta

The type II error.

n

The total sample size.

ncovariates

The number of covariates.

nconfigs

The number of configurations of discretized covariate values.

x

The matrix of covariate values.

pconfigs

The vector of probabilities for the configurations.

corr

The multiple correlation between the predictor and other covariates. Defaults to 0.

oddsratios

The odds ratios for one unit increase in the covariates.

responseprob

The response probability in the full model when all predictor variables are equal to their means.

rounding

Whether to round up sample size. Defaults to 1 for sample size rounding.

alpha

The two-sided significance level. Defaults to 0.05.

Details

We consider the logistic regression of a binary response variable Y on a set of predictor variables x = (x_1,\ldots,x_K)^T with x_1 being the covariate of interest: \log \frac{P(Y_i=1)}{1 - P(Y_i = 1)} = \psi_0 + x_i^T \psi, where \psi = (\psi_1,\ldots,\psi_K)^T. Similar to Self et al (1992), we assume that all covariates are either inherently discrete or discretized from continuous distributions (e.g. using the quantiles). Let m denote the total number of configurations of the covariate values. Let

\pi_i = P(x = x_i), i = 1,\ldots, m

denote the probabilities for the configurations of the covariates under independence. The likelihood ratio test statistic for testing H_0: \psi_1 = 0 can be approximated by a noncentral chi-square distribution with one degree of freedom and noncentrality parameter

\Delta = 2 \sum_{i=1}^m \pi_i [b'(\theta_i)(\theta_i - \theta_i^*) - \{b(\theta_i) - b(\theta_i^*)\}],

where

\theta_i = \psi_0 + \sum_{j=1}^{k} \psi_j x_{ij},

\theta_i^* = \psi_0^* + \sum_{j=2}^{k} \psi_j^* x_{ij},

for \psi_0^* = \psi_0 + \psi_1 \mu_1, and \psi_j^* = \psi_j for j=2,\ldots,K. Here \mu_1 is the mean of x_1, e.g., \mu_1 = \sum_i \pi_i x_{i1}. In addition, by formulating the logistic regression in the framework of generalized linear models,

b(\theta) = \log(1 + \exp(\theta)),

and

b'(\theta) = \frac{\exp(\theta)}{1 + \exp(\theta)}.

The regression coefficients \psi can be obtained by taking the log of the odds ratios for the covariates. The intercept \psi_0 can be derived as

\psi_0 = \log(\bar{\mu}/(1- \bar{\mu})) - \sum_{j=1}^{K} \psi_j \mu_j,

where \bar{\mu} denotes the response probability when all predictor variables are equal to their means.

Finally, let \rho denote the multiple correlation between the predictor and other covariates. The noncentrality parameter of the chi-square test is adjusted downward by multiplying by 1-\rho^2.

Value

An S3 class designLogistic object with the following components:

• power: The power to reject the null hypothesis.

• alpha: The two-sided significance level.

• n: The total sample size.

• ncovariates: The number of covariates.

• nconfigs: The number of configurations of discretized covariate values.

• x: The matrix of covariate values.

• pconfigs: The vector of probabilities for the configurations.

• corr: The multiple correlation between the predictor and other covariates.

• oddsratios: The odds ratios for one unit increase in the covariates.

• responseprob: The response probability in the full model when all predictor variables are equal to their means.

• effectsize: The effect size for the chi-square test.

• rounding: Whether to round up sample size.

Author(s)

Kaifeng Lu, [e-mail]

References

Steven G. Self, Robert H. Mauritsen and Jill Ohara. Power calculations for likelihood ratio tests in generalized linear models. Biometrics 1992; 48:31-39.

Examples

# two ordinal covariates
x1 = c(5, 10, 15, 20)
px1 = c(0.2, 0.3, 0.3, 0.2)

x2 = c(2, 4, 6)
px2 = c(0.4, 0.4, 0.2)

# discretizing a normal distribution with mean 4 and standard deviation 2
nbins = 10
x3 = qnorm(((1:nbins) - 0.5)/nbins)*2 + 4
px3 = rep(1/nbins, nbins)

# combination of covariate values
nconfigs = length(x1)*length(x2)*length(x3)
x = expand.grid(x3 = x3, x2 = x2, x1 = x1)
x = as.matrix(x[, ncol(x):1])

# probabilities for the covariate configurations under independence
pconfigs = as.numeric(px1 %x% px2 %x% px3)

# convert the odds ratio for the predictor variable in 5-unit change
# to the odds ratio in 1-unit change
(design1 <- getDesignLogistic(
  beta = 0.1, ncovariates = 3,
  nconfigs = nconfigs,
  x = x,
  pconfigs = pconfigs,
  oddsratios = c(1.2^(1/5), 1.4, 1.3),
  responseprob = 0.25,
  alpha = 0.1))