Prevalence ratio estimation via logistic regression: a tool in R

Abstract

The interpretation of odds ratios (OR) as prevalence ratios (PR) in cross-sectional studies have been criticized since this equivalence is not true unless under specific circumstances. The logistic regression model is a very well known statistical tool for analysis of binary outcomes and frequently used to obtain adjusted OR. Here, we introduce the prLogistic for the R statistical computing environment which can be obtained from The Comprehensive R Archive Network, https://cran.r-project.org/package=prLogistic. The package prLogistic was built to assist the estimation of PR via logistic regression models adjusted by delta method and bootstrap for analysis of independent and correlated binary data. Two applications are presented to illustrate its use for analysis of independent observations and data from clustered studies.

Key words
Logistic model; delta method; bootstrap; prevalence ratios

INTRODUCTION

The concept of risk is fundamental in several research areas, being the measures of risk associated to the probability of occurrence of an event of interest. Particularly in Public Health and Epidemiology, two commonly used measures to estimate risk are relative risk (RR), in longitudinal studies, and prevalence ratios (PR), in cross-sectional studies. In the simplest situation, unadjusted measures can be computed easily throughout analysis of contingency tables. Another measure of association frequently reported by epidemiologists and medical researchers is the odds ratio (OR), which differs mathematically from RR and PR. It is well known that OR overestimates relative risk (RR) and prevalence ratios (PR) when the event is not rare (Stromberg 1994, Thompson et al. 1998, McNutt et al. 2003, Greenland 2004, Newcombe 2006, Tamhane et al. 2016).

When the main interest of the investigator relies on the estimation of an association or risk measure adjusted by covariates or confounders, use of statistical modeling is usually required. In epidemiology, several outcomes are binary and logistic regression models are widely applied. Using logistic regression models, one can easily estimate OR $=exp\left(\beta \right)$, where $\beta$ is a parameter related to the risk factor of interest.

However, interpretation of OR as a risk measure might be misleading in terms of how it can be interpreted. Many researchers mistakenly interpret odds as risk even when OR provides a poor approximation to PR. Phrases including terms like risk, "likelihood", "probability" and “more likely" to interpret the OR are commonly found in the literature. In certain circumstances it is possible to estimate RR or PR through their relationships to OR. Nevertheless, the computation of the corresponding confidence intervals is not trivial and some unsuccessful methods had been proposed (Zhang & Yu 1998).

There is some debate in the literature about alternative approaches to obtain adjusted measures of PR in cross-sectional studies (Barros & Hirakata 2003, Localio et al. 2007, Petersen & Deddens 2008, Cummings 2009, Savu et al. 2010). One of the proposals is to estimate PR using logistic regression (Oliveira et al. 1997, Localio et al. 2007). A more recent discussion is about how to estimate adjusted PR for correlated data, particularly in the analysis of clustered data (Bastos et al. 2015, Santos et al. 2008).

Implementation of new statistical methods and its availability for applied researchers is other concern among data analysts. Many of the most recently proposed statistical methods can not be applied to data analysis because they are not easily accessible via standard statistical software.

Here, we introduce prLogistic, an R package specifically built to assist estimation of PRs in cross-sectional studies via logistic regression models for analysis of both independent and correlated data. prLogistic currently contains three main functions. The first one allows estimates PR using logistic models. The second function performs PR estimation using logistic models with conditional standardization. Finally, the third function estimates PR using a logistic model with marginal standardization. We provide a “how to“ guide to use those functions by applying them to empirical data sets and supply insights on interpreting the outputs.

This paper is structured as follows. In Section 2 we outline the theory underlying PR estimation via logistic models while in Section 3 we describe the functions contained in our prLogistic package. We provide two empirical applications to illustrate the use of the prLogistic package in Section 4. Finally, Section 5 contains concluding remarks and directions for future research.

1 - ESTIMATION OF PREVALENCE RATIOS

The OR can be defined by the ratio of two odds, such that OR $=\left(p_1/(1-p_1)\right)/\left(p_0/(1-p_0)\right)$, where $p_1$ and $p_0$ denote, respectively, the prevalence of the event of interest in the exposed and non-exposed groups. The PR, on the other hand, is defined by the ratio of two proportions given by $$PR=p_1/p_0$$. Therefore, interpretation of these two measures is not the same, unless the event is rare, which implies $(1-p_1)\to 1$ and $(1-p_0)\to 1$.

Let $Y$ be the binary outcome, where $Y=1$ if the outcome is a “success“, whatever your definition, and $Y=0$ otherwise. The probability of success is assumed to depend on known covariates, i.e., we consider the logistic regression model

$$𝖤(Y|𝐗)=P(Y=1|𝐗)=\frac{exp\left(𝐗\beta \right)}{1+exp\left(𝐗\beta \right)},$$

where $𝐗=[X_1,X_2,...,X_k]$ is a matrix of independent variables and $\beta ={({\beta }_0,\dots ,{\beta }_k)}^{\top }$ is a vector of model parameters.

We are interested in obtaining an expression for estimating PR as a function of $\beta .$ For instance, suppose that we are evaluating the effect of a binary exposure $X_1$ (0/1) on the occurrence of $Y$ after adjustment by $k-1$ independent variables ($X_2,...,X_k$). In this case, PR is given by

$$\mathrm{P}\mathrm{R}=\frac{1+\exp \{-{\beta }_0-{\beta }_2{X}_2-\cdots -{\beta }_k{X}_k\}}{1+\exp \{-{\beta }_0-{\beta }_1-{\beta }_2{X}_2-\cdots -{\beta }_k{X}_k\}}.$$

Note that, in this expression, PR is also function of the values of the independent variables included in the model, differently from OR.

1.1 - Standardization Procedures

Some standardization procedures for effect measures based on regression models had been proposed in the literature (Wilcosky & Chambless 1985, Lane & Nelder 1982), being the two most commonly used methods called conditional and marginal standardization. For the conditional standardization procedure, a reference or baseline value (for instance, the mean for continuous variables) of each variable included in the model is defined and, thus, the prevalence for each group ($X_1=1$ e $X_1=0$) is calculated. Using conditional standardization via logistic regression, the investigator is interested in finding the PR comparing exposure status (present/absent, i.e., $X_1=1$/$X_1=0$) at a fixed level of covariates.

For the marginal standardization procedure, on the other hand, the prevalence is computed, for each group (say $p_{1i}$ and $p_{0i}$, respectively, for exposed and non exposed groups) using the individual values for the covariates and later getting the average value among all observations. Therefore, the marginal standardized prevalences for the exposed and non exposed subjects are given, respectively, by averaging $p_{1i}$ and $p_{0i}$ across all $i$ subjects.

As an example consider data on $n$ subjects with a binary exposure $X_1$ (1 = exposed, 0 = non-exposed), and a continuous variable $X_2$. Using the conditional standardization procedure, the adjusted PR is given by

$$\mathrm{P}\mathrm{R}=\frac{1+\exp \{-{\beta }_0-{\beta }_2{\overline{X}}_2)}{1+\exp \{-{\beta }_0-{\beta }_1-{\beta }_2{\overline{X}}_2\}},$$

where ${\overline{X}}_2$ denotes the mean of $X_2$. If $X_2$ were a binary covariate, the researcher should set specific value for computing PR (for instance $X_2=0$).

For the marginal method the adjusted PR is defined by

$$\mathrm{P}\mathrm{R}=\frac{{\displaystyle \frac{1}{n}\sum _i(1/\{1+\exp (-({\beta }_0+{\beta }_1+{\beta }_2{X}_{2i}))\})}}{{\displaystyle \frac{1}{n}\sum _i(1/\{1+\exp (-({\beta }_0+{\beta }_2{X}_{2i}))\})}},$$

where the sum is over all $n$ subjects.

1.2 - Inference for Prevalence Ratios

Methods for obtaining confidence intervals for PR include delta method and bootstrap. The delta method is a general technique for asymptotic distribution of random variable functions that is based on approximation by Taylor series (Bishop et al. 2007). Let $(X_1,X_2,...,X_k)$ be a $k$-dimensional random vector and $h(X_1,X_2,...,X_k)$ be a function defined on an open subset of $k$-dimensional space to real values. We assume that $h(\cdot )$ is differential and $𝖤\left(X_i\right)={\mu }_i$. Thus

$$𝖵𝖠𝖱\left(h(X_1,X_2,...,X_k)\right)\approx \sum _{i=1}^k{\left(\frac{\partial h}{\partial {\mu }_i}\right)}^2\cdot 𝖵𝖠𝖱\left(X_i\right)+2\sum \sum _{ij}^k\frac{{\partial }^{2}h}{\partial {\mu }_i{\mu }_j}𝖢𝖮𝖵(X_i,X_j),$$

which involves partial derivates of the function of interest. For the estimation of the variance of PR, log(PR) is asymptotically normally distributed and we use the delta method for estimating VAR(log(PR)), where

$$\widehat{\mathsf{V}\mathsf{A}\mathsf{R}}(\log (\widehat{\mathrm{P}\mathrm{R}}))\approx X_{\ast }\hat{\mathrm{\Sigma }}{X}_{\ast }^{{}^{\prime }},$$

with $X_{*}={\hat{q}}_1{X}_1-{\hat{q}}_0{X}_0$, ${\hat{q}}_1=1-{\hat{p}}_1$, ${\hat{q}}_0=1-{\hat{p}}_0$, $X_1=[1,1,X_2,...,X_k]$, $X_0=[1,0,X_2,...,X_k]$, and $\hat{\Sigma }$ is the covariance matrix of the model parameters ($\beta$’s) (Oliveira 1997). Using the delta method, the adjusted $(1-\alpha )\%$ confidence intervals (CIs) for PR are defined by

$$\exp (\log (\widehat{\mathrm{P}\mathrm{R}})\pm z_{\alpha /2}\widehat{\mathrm{s}\mathrm{e}}(\log (\widehat{\mathrm{P}\mathrm{R}}))),$$

where $$\log (\widehat{\mathrm{P}\mathrm{R}})$$ is the estimate for adjusted $$\log (\mathrm{P}\mathrm{R})$$, $$\widehat{\mathrm{s}\mathrm{e}}(\log (\widehat{\mathrm{P}\mathrm{R}}))$$ is the estimate of standard-error for $$\log (\mathrm{P}\mathrm{R})$$ and $z_{\alpha /2}$ is the quantile of a standard normal distribution, $\alpha$ being the significance level.

The bootstrap approach, in its turn, is based on resampling with replacement for estimation of functions of interest (Efron & Tibshirani 1993). For instance, we can consider 1,000 bootstrap replicates to produce a bootstrap distribution of PR values. Using bootstrap estimates for sample variance (Davison & Hinkley 1997), the first order normal approximation bootstrap $(1-\alpha )\%$ CI is

$$\exp (\log (\widehat{{\mathrm{P}\mathrm{R}}^{\ast }})\pm z_{\alpha /2}\widehat{{\mathrm{s}\mathrm{e}}^{\ast }}(\log (\widehat{{\mathrm{P}\mathrm{R}}^{\ast }}))),$$

where $$\log (\widehat{{\mathrm{P}\mathrm{R}}^{\ast }})$$ is the bootstrap estimate for adjusted $$\log (\mathrm{P}\mathrm{R})$$ and $$\widehat{{\mathrm{s}\mathrm{e}}^{\ast }}(\log (\widehat{{\mathrm{P}\mathrm{R}}^{\ast }}))$$ is the bootstrap estimate for the standard error of $$\log (\widehat{\mathrm{P}\mathrm{R}})$$. An alternative approach, using bootstrap percentile interval (Fox 2015), considers the empirical quantiles of bootstrap estimates for defining the interval. In such situation, the interval limits are given, for instance, by percentiles 2.5 and 97.5 when we are interested in the 95$\%$ CI.

1.3 - Random-Effects Logistic Model

Random-effect models, also known as mixed models or multilevel models, are often used for modeling correlated data (Diggle et al. 1994, Hox et al. 2017). Such data arises from the sampling design, including the use of cluster sampling, where individuals are nested in geographic areas or institutions, such as schools or companies, and the use of longitudinal studies, which investigates changes in repeated measures of the outcome over time for the same sampling unit. These models can be applied for analysis of a variety of outcomes types: continuous, binary, polytomous, counts, time-to-event, etc. Here we focus on the analysis of binary outcomes. These models make adjustments for non-observed individual characteristics, which reflect a natural heterogeneity among subjects. Let $Y_{ij}$ be the binary outcome variable at cluster $j$ for subject $i$, and denote $X_1$ and $X_2$ two independent variables. The random-effects logistic model can be defined by

$$\mathrm{logit}[\mathsf{P}(Y_{ij}|X_{1ij},X_{2ij},u_{oj})]={\beta }_0+{\beta }_1{X}_{1ij}+{\beta }_2{X}_{2ij}+u_{oj},$$

where $u_{oj}\sim N(0,{\sigma }^2)$ represents a cluster specific random effect. Using the estimates for the parameters of this model ($\beta$’s), we can obtain PR as defined previously. Interpretation of regression coefficients from the random-effects logistic model has to be done by conditioning on the random effects (Larsen et al. 2000, McCulloch & Searle 2001).

2 - THE R PACKAGE PRLOGISTIC

The prLogistic package is implemented under the FLOSS (Free/Libre Open Source Software) paradigm in the R system for statistical computing (R Development Core Team 2021) and it is available from the Comprehensive R Archive Network (CRAN) at https://cran.r-project.org/package=prLogistic and the experimental updates at GitHub repository https://github.com/Raydonal/prLogistic. It takes advantage of functionality developed in other packages as epiR (Stevenson et al. 2013), boot (Canty 2002) and lme4 (Bates et al. 2015, 2021).

The main functionalities of the R package prLogistic (Ospina & Amorim 2021) are:

• The function prLogisticDelta estimates prevalence ratios (PRs) and their confidence intervals using logistic models. The estimation of standard errors for PRs is done using the delta method. The function prLogisticDelta allows estimation of PRs using two standardization procedures: conditional or marginal (Wilcosky Chambless 1985).

  A typical form used with glm() function is included in the formula argument as response terms, where the response is the (binary) response vector and terms is a series of variables which specifies a linear predictor for the response. The prLogisticDelta assumes a binomial family associated with it. The lmer() function is used when a vertical bar character | separates an expression for a model matrix and a grouping factor. The output returned by prLogisticDelta contains prevalence ratio and its 95% confidence intervals.

• The function prLogisticBootCond() estimates prevalence ratios (PRs) and bootstrap confidence intervals using logistic models with conditional standardization. The estimation of standard errors for PRs is given through bootstrapping. The fitted model object can be obtained using glm() function for binary responses when unit samples are independent. The lmer() function should be used for correlated binary responses. The output returned by prLogisticBootCond contains prevalence ratios and their 95% bootstrap confidence intervals with conditional standardization. Both normal and percentile bootstrap confidence intervals are presented.

• The function prLogisticBootMarg() estimates prevalence ratios (PRs) and bootstrap confidence intervals using logistic models with marginal standardization. The estimation of standard errors for PRs is given through bootstrapping. The fitted model object can be obtained using glm() function for binary responses when unit samples are independent. The lmer() function should be used for correlated binary responses. The output returned by prLogisticBootMarg contains prevalence ratios and their 95% bootstrap confidence intervals with marginal standardization. Both normal and percentile bootstrap confidence intervals are presented.

For the functions prLogisticDelta, prLogisticBootCond and prLogisticBootMarg, confidence intervals of $(1-\alpha )$% for PRs are available for standard logistic regression and for random-effects logistic models (Santos et al. 2008). If categorization for predictors is other than (0,1), factor() should be considered.

3 - APPLICATIONS

We illustrate the use of the R implemented functions prLogisticDelta(), prLogisticBootCond() and prLogisticBootMarg(), which are available in prLogistic package. We describe two datasets that are used in the examples. The first example is related to data from randomized clinical trials to evaluate the impact of intervention programs on drug use reduction (dataset UIS), which contains independent observations. The second example, on the other hand, is an observational clustered study about primary education in Thailand (dataset Thailand).

3.1 - The Umaru Impact Study

Dataset UIS contains information from randomized trials related to treatment for drug abuse obtained by the University of Massachusetts Aids Research Unit (UMARU) IMPACT Study (UIS). The study aimed to compare treatment programs of different durations in the reduction of drug abuse and in the prevention of high-risk HIV behavior. The variables on the dataset available at prLogistic package are age at enrollment, intravenous (IV) drug use history at admission, race, treatment group, treatment site, and patient’s status at the end of the treatment program (Hosmer Jr et al. 2013).

We load the package prLogistic and dataset UIS, and look at the first 5 rows of the data:

Dataset UIS contains the following subset of the variables from the original study:

• ID is the patient identification code.

• Age is the age at enrollment (in years) recoded to 1 = 32 years or younger; 0 = otherwise.

• DrugUse is the IV drug use history at admission (1 = never; 0 = previous or recent).

• race is the patient’s race (1 = other; 0 = white).

• trt is the treatment group (1 = long; 0 = short).

• site is the treatment site (1 = B; 0 = A).

• drugFree is an indicator of returning to drug use prior to the scheduled end of the treatment program (1 = remained drug free; 0 = otherwise).

We describe the outcome variable using the following:

We noted that about 26% of the patients remained drug free at the end of the treatment program. The description of the outcome according to treatment group is given by:

Since the outcome is relatively common, ORs do not approximate prevalence ratios. We fit the following logistic regression model

$$\log \left\{\frac{\mathsf{P}(Y_i=1)}{\mathsf{P}(Y_i=0)}\right\}={\beta }_0+{\beta }_1{\mathtt{Age}}_i+{\beta }_2{\mathtt{DrugUse}}_i+{\beta }_3{\mathtt{trt}}_i.$$

We estimate the prevalence ratios (PRs) in model ($2$) with 95% confidence intervals using delta method and conditional standardization through:

For comparison, note that the odds ratios can be estimated using logistic regression as:

Note that both ORs and PRs were estimated using logistic regression. Considering the previous results, we can observe, for instance, that patients who never used IV drugs before admission were more likely to remain drug free than patients with previous/recent IV drug use history ($$\widehat{\mathrm{P}\mathrm{R}}$$ = 1.84 (95% CI: 1.38; 2.44); $$\widehat{\mathrm{O}\mathrm{R}}$$ = 2.32 (95% CI: 1.54; 3.48)).

We can obtain 95% bootstrap confidence intervals for PR with conditional standardization using:

Note that function prLogisticBootCond provides two bootstrap confidence intervals: (a) one based on normal theory, which is often approximately the case in sufficiently large samples, (b) the other using empirical quantiles of the bootstrap estimates to form the interval.

Based on the results of the conditional standardization, the probability of remaining drug-free by the end of the treatment program is $42\%$ larger for those participating in the long treatment group compared to those in the short group among patients with more than 32 years old and who used IV drug previously or recently before to admission.

Similarly, we can estimate PRs using marginal standardization with delta method:

or with bootstrap confidence intervals:

Considering the results with marginal standardization (population-averaged), the probability of remaining drug free at the end of the treatment program, assuming that all patients were in the long group, is $39\%$ larger than that same probability when all patients were assumed to be in the short group. For comparison purposes, we plot these estimates (see Figure 1).

Figure 1
Marginal Prevalence Ratios (PR) and corresponding 95% CIs defined by three methods: (a) Delta; (b) Bootstrap-Normal approximation; (c) Boostrap-Percentile Interval.

Note that $95\%$ CIs are very similar for the three methods for estimating marginal prevalence ratios in this example (see Figure 1). The major difference is that statistical significance is not reached when using bootstrap with Normal approximation.

3.2 - Education in Thailand

Data are from a large national survey on primary education in Thailand, including information for 8,582 sixth graders nested within 411 schools (Raudenbush & Bhumirat 1992). The binary outcome variable “rgi" indicates whether a student has repeated a grade during primary education. The predictor variables in the dataset are the child’s sex and the child’s pre-primary education. Every level-1 record corresponds to a student. The level-2 is defined by schools.

We load the prLogistic package and the dataset Thailand, and explore the dataset:

Dataset Thailand contains the following variables from the original study:

• schoolid is the school identification.

• sex is the student’s sex (1 = boy, 0 = girl).

• ped is an indicator for pre-primary education (1 = yes, 0 = no).

• rgi is an indicator whether a student has ever repeated a class (1 = yes, 0 = no).

The distribution of the outcome variable (repeated grade indicator – rgi) is given by:

Due to clustering, the following random-effects logistic model was fitted:

$$\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}[\mathsf{P}({\mathtt{rgi}}_{ij}|{\mathtt{sex}}_{ij},{\mathtt{pped}}_{ij},u_{oj})]={\beta }_0+{\beta }_1{\mathtt{sex}}_{ij}+{\beta }_2{\mathtt{pped}}_{ij}+u_{oj}.$$

Using a random intercept logistic model without covariates, the intraclass correlation coefficient (ICC) is 0.33, indicating an important effect of clustering that has to be considered in the analysis. Thus, the random effects logistic model is indicated for this data analysis instead of traditional logistic regression, which assumes the independence of observations.

The conditional PR using delta method can be obtained by adding the option cluster="TRUE" in the function prLogisticDelta as follows:

To get the marginal estimates using this function, we should also include the option pattern="marginal" as showed below:

We also can obtain estimates for a different confidence level. For instance, suppose we are interested in computing a 90% confidence interval for PR. In this case we use

For obtaining bootstrap confidence intervals for PR we use

or

respectively, for conditional and marginal standardization. Alternatively, we could obtain these estimates using the following syntax:

Similar syntax could be used with function prLogisticBootMarg to obtain estimates with marginal standardization.

We could also modify the number of bootstrap replications and the confidence level using:

Both predictors are significantly associated to whether a student has repeated a grade during primary education, i.e., in a given school the boys have 63% higher probability of repetition than girls (PR = 1.63 [95% CI = 1.49; 1.92]) and a child has 43% less probability if he/she received pre-primary education (PR = 0.57 [95 %CI = 0.45; 0.62]). These results were based on marginal standardization procedure with bootstrap-percentile 95% confidence interval. Similar conclusions were reached by using any of the methods described here.

4 - CONCLUSION AND FUTURE WORK

We have shown how logistic regression models can be implemented to estimate the prevalence ratios and their confidence intervals using our prLogistic package, in situations where the observations are independent or when data comes from clustered studies. The package can accommodate the information of conditional and marginal standardization, commonly used in epidemiology, as well as either delta method or bootstrap resampling for the obtention of confidence intervals. Our package is easily used and does not involve extensive programming. Our contribution of the package prLogistic will make these methodologies more accessible to applied researchers. In future updates of the package, the functions will implement generalized estimating equations (GEE), a marginal approach for longitudinal/clustered data. It is, however, worth mentioning that most analysis in Epidemiology and Public Health involves only categorical variables. Future implementation might consider the extension of the procedures to incorporate other types of variables, nonlinear and interaction terms between covariates and include the sampling design via the R package survey (Lumley 2004, Oberski 2014).

ACKNOWLEDGMENTS

This work was supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) under Grant 305305/2019-0 and Fundação de Amparo à Pesquisa do Estado da Bahia (FAPESB) /Brazil. We are also thankful to two anonymous referees whose comments and suggestions led to a much improved manuscript.

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Publication Dates

• Publication in this collection
  17 Sept 2021
• Date of issue
  2021

History

• Received
  21 Mar 2019
• Accepted
  11 July 2019