Let Continuous Outcome Variables Remain Continuous

Abstract

The complementary log-log is an alternative to logistic model. In many areas of research, the outcome data are continuous. We aim to provide a procedure that allows the researcher to estimate the coefficients of the complementary log-log model without dichotomizing and without loss of information. We show that the sample size required for a specific power of the proposed approach is substantially smaller than the dichotomizing method. We find that estimators derived from proposed method are consistently more efficient than dichotomizing method. To illustrate the use of proposed method, we employ the data arising from the NHSI.

1. Introduction

Recently, logistic regression has become a popular tool in biomedical studies. The parameter in logistic regression has the interpretation of log odds ratio, which is easy for people such as physicians to understand. Probit and complementary log-log are alternatives to logistic model. For a covariate 𝑋 and a binary response variable 𝑌, let 𝜋(𝑋)=𝑃(𝑌=1∣𝑋=𝑥). A related model to the complementary log-log link is the log-log link. For it, 𝜋(𝑥) approaches 0 sharply but approaches 1 slowly. When the complementary log-log model holds for the probability of a success, the log-log model holds for the probability of a failure [1].

These models use a categorical (dichotomous or polytomous) outcome variable. In many areas of research, the outcome data are continuous. Many researchers have no hesitation in dichotomizing a continuous variable, but this practice does not make use of within-category information. Several investigators have noted the disadvantages of dichotomizing both independent and outcome variables [2–10]. Ragland [11] showed that the magnitude of odds ratio and statistical power depend on the cutpoint used to dichotomize the response variable. From a clinical point of view, binary outcomes may be preferred for some reasons such as (1) setting diagnostic criteria for disease, (2) offering a simpler interpretation of common effect measures from statistical models such as odds ratios and relative risks. However, all advantages come at the lost information. From a statistical point of view, this loss of information means more samples which are required to attain prespecified powers.

Moser and Coombs [12] provided a closed-form relationship that allows a direct comparison between the logistic and linear regression coefficients. They also provided a procedure that allows the researcher to analyze the original continuous outcome without dichotomizing. To date, a method that applies the complementary log-log model without dichotomizing and without loss of information has not been available.

We aim to (a) provide a method that allows the researcher to estimate the coefficients of the complementary log-log model without dichotomizing and without loss of information, (b) show that the coefficient of the complementary log-log model can be interpreted in terms of the regression coefficients, (c) demonstrate that the coefficient estimates from this method have smaller variances and shorter confidence intervals than the dichotomizing method.

2. Methods

2.1. Model

Let 𝑦₁,𝑦₂,…,𝑦_𝑛 be 𝑛 independent observations on 𝑦, and let 𝑥₁,𝑥₂,…,𝑥_𝑝−1 be 𝑝−1 predictor variables thought to be related to the response variable 𝑦. The multiple linear regression model for the 𝑖th observation can be expressed as𝑦_𝑖=𝛽₀+𝛽₁𝑥_𝑖1+𝛽₂𝑥_𝑖2+⋯+𝛽_𝑝−1𝑥_𝑖𝑝−1+𝐸_𝑖𝑖=1,2,…,𝑛, (1) or𝑦_𝑖=𝑥_𝑖𝛽+𝐸_𝑖𝑖=1,2,…,𝑛, (2) where𝑥_𝑖=1,𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑝−1. (3) To complete the model, we make the following assumptions:(1)𝐸(𝐸_𝑖)=0 for 𝑖=1,2,…,𝑛,(2)var(𝐸_𝑖)=𝜎² for 𝑖=1,2,…,𝑛,(3)the independent 𝐸_𝑖 follows an extreme value distribution for 𝑖=1,2,…,𝑛.

Writing the model for each of the 𝑛 observations, in matrix form, we have𝑦₁𝑦₂..𝑦_𝑛=1𝑥₁₁𝑥₁₂…𝑥_1𝑝−11𝑥₂₁𝑥₂₂…𝑥_2𝑝−1..1𝑥_𝑛1𝑥_𝑛2…𝑥_𝑛𝑝−1𝛽₀𝛽₁..𝛽_𝑝−1+𝐸₁𝐸₂..𝐸_𝑛, (4) or𝑦=𝑋𝛽+𝐸. (5) The preceding three assumptions on 𝐸_𝑖 and 𝑦_𝑖can be expressed in terms of this model:(1)𝐸(𝐸)=0, (2)cov(𝐸)=𝜎²𝐼, (3)the 𝐸_𝑖 is extreme value (0,𝜎²) for 𝑖=1,2,…,𝑛.

2.2. (Largest) Extreme Value Distribution

The PDF and CDF of the extreme value distribution are given by 𝑓=𝜋𝑦∣𝑥𝛽,𝜎𝜎√6×exp−𝑦−𝑥𝛽−𝑘𝜎𝜎×𝜋√6−exp𝑦−𝑥𝛽−𝑘𝜎𝜎×𝜋√6𝑃−−∞⟨𝑥⟨∞,𝜎⟩0,𝑦≤𝑐=exp−exp𝑐−𝑥𝛽+𝑘𝜎𝜎×𝜋√6−∞⟨𝑥⟨∞,𝜎⟩,𝑘≈0.45. (6)

It is easy to check that𝜔_𝑗=ln𝜋₁ln𝜋₂=ln𝑝𝑦≤𝑐∣𝑥ln𝑝𝑦≤𝑐∣𝑥_(−1,𝑗)=−exp−𝑐−𝑥^′𝛽+𝑘𝜎/𝜎×𝜋/√6−exp−𝑐−𝑥^′_−1,𝑗𝛽+𝑘𝜎/𝜎×𝜋/√6𝜋=exp√6⋅𝛽_𝑗𝜎⟹𝜋₁=𝜋₂^{√exp((𝜋/6)⋅(𝛽𝑗/𝜎))}, (7) where𝑥=1,𝑥₁,…,𝑥_𝑗,…,𝑥_𝑝−1,𝑥_(−1,𝑗)=1,𝑥₁,…,𝑥_𝑗−1,…,𝑥_𝑝−1,𝛽=𝛽₀,𝛽₁,…,𝛽_𝑗,…,𝛽_𝑝−1^′. (8) To return to a random sample of observations (𝑦₁,𝑦₂,…,𝑦_𝑛), we conclude that the PDF and CDF of each independent 𝑦_𝑖 are given by (6), and the corresponding equality (7) is given byln𝜋₁ln𝜋₂𝜋=exp𝜎√6𝛽_𝑗, (9) where the estimate 𝛽_𝑗is the (𝑗+1)th element of vector 𝛽=(𝛽₀,𝛽₁,…,𝛽_𝑗,…,𝛽_𝑝−1)^′. It is readily shown that the results also hold true for the smallest extreme value distribution (Appendix A).

2.3. The Proposed Confidence Intervals

Let𝛽=𝛽₀,𝛽₁,…,𝛽_𝑗,…,𝛽_𝑝−1^′=𝑋^′𝑋⁻¹𝑋^′𝜎𝑌𝑗=0,…,𝑝−1,₂=𝑌^′𝐼_𝑛−𝑋𝑋^′𝑋⁻¹𝑋^′𝑌.𝑛−𝑝 (10) According to the preceding three assumptions on 𝐸_𝑖 and𝑦_𝑖, we obtain𝐸𝛽=𝐸𝑋^′𝑋⁻¹𝑋^′𝑌=𝑋^′𝑋⁻¹𝑋^′𝐸𝑌=𝑋^′𝑋⁻¹𝑋^′𝐸𝜎𝑋𝛽=𝛽,₂=1𝐸𝑌𝑛−𝑝^′𝐼_𝑛−𝑋𝑋^′𝑋⁻¹𝑋^′𝑌=1𝐼𝑛−𝑝tr_𝑛−𝑋𝑋^′𝑋⁻¹𝑋^′𝜎²𝐼+𝐸𝑌^′𝐼_𝑛−𝑋𝑋^′𝑋⁻¹𝑋^′𝐸(𝑌=1)𝜎𝑛−𝑝²tr𝐼_𝑛−𝑋𝑋^′𝑋⁻¹𝑋^′+𝛽^′𝑋^′𝐼_𝑛−𝑋𝑋^′𝑋⁻¹𝑋^′=1𝑋𝛽𝜎𝑛−𝑝²𝑛−tr𝑋𝑋^′𝑋⁻¹𝑋^′+𝛽^′𝑋^′𝑋𝛽−𝛽^′𝑋^′𝑋𝑋^′𝑋⁻¹𝑋^′=1𝑋𝛽𝜎𝑛−𝑝²𝑛−tr𝑋𝑋^′𝑋⁻¹𝑋^′+𝛽^′𝑋^′𝑋𝛽−𝛽^′𝑋^′=1𝑋𝛽𝜎𝑛−𝑝²𝑛−tr𝐼_𝑃=1𝜎𝑛−𝑝²𝑛−𝑝=𝜎². (11) Therefore, 𝛽 and𝜎²are unbiased estimators of 𝛽 and𝜎².

We have assumed that 𝐸_𝑖 is distributed as an extreme value, and we use the approximation of the extreme value distribution of the errors 𝐸_𝑖 by the normal distribution. For normally distributed observations, 𝛽_𝑗/(𝜎𝛿_𝑗) follows a noncentral 𝑡 distribution with 𝑛−𝑝 degree of freedom and noncentrality parameter −∞<𝛽_𝑗/(𝜎𝛿_𝑗)<∞,1−𝛼=𝑃𝑡_1−(𝛼/2)𝛽𝑛−𝑝,^𝑗𝜎𝛿_𝑗<𝛽_𝑗𝜎𝛿_𝑗<𝑡_𝛼/2𝛽𝑛−𝑝,^𝑗𝜎𝛿_𝑗, (12) where 𝑡_𝛼/2[𝑟,𝑠] represents the 100(1−(𝛼/2))percentile point of a noncentral 𝑡 distribution with 𝑟 degrees of freedom and noncentrality parameter −∞<𝑠<∞, and 𝛿_𝑗 is the (𝑗+1)st diagonal element of (𝑋^′𝑋)⁻¹. We use the approximation of the percentiles of the noncentral 𝑡 distribution by the standard normal percentiles [13], then𝛽1−𝛼=𝑃^𝑗/𝜎𝛿_𝑗−𝑧_𝛼/21+𝛽²_𝑗/𝜎²𝛿_𝑗−𝑧²_𝛼/2/2𝑛−𝑝^1/21−𝑧²_𝛼/2<𝛽/2𝑛−𝑝_𝑗𝜎𝛿_𝑗<𝛽_𝑗/𝜎𝛿_𝑗+𝑧_𝛼/21+𝛽²_𝑗/𝜎²𝛿_𝑗−𝑧²_𝛼/2/2𝑛−𝑝^1/21−𝑧²_𝛼/2,𝛽/2𝑛−𝑝_𝑗𝜎^𝑈=𝛽_𝑗𝜎𝑧1−^2𝛼/22𝑛−𝑝+𝑧_𝛼/2𝛿_𝑗𝛽1+^2𝑗/𝜎²𝛿_𝑗−𝑧²_𝛼/22𝑛−𝑝^1/2,𝛽_𝑗𝜎^𝐿=𝛽_𝑗𝜎𝑧1−^2𝛼/22𝑛−𝑝−𝑧_𝛼/2𝛿_𝑗𝛽1+^2𝑗/𝜎²𝛿_𝑗−𝑧²_𝛼/22𝑛−𝑝^1/2, (13)Thus, we obtain an approximate 100(1−𝛼) percent confidence interval for 𝜔_𝑗𝜋exp√6𝛽_𝑗𝜎^𝐿𝜋,exp√6𝛽_𝑗𝜎^𝑈. (14)

3. Comparison of the Two Methods

Let 𝑌_𝑖 be a continuous outcome variable. For fixed value of 𝐶, we define 𝑌^∗_𝑖 such that𝑌^∗_𝑖=1if𝑌_𝑖≥𝐶,0if𝑌_𝑖<𝐶. (15) Suppose that 𝑌^∗₁,…,𝑌^∗_𝑛 form a random sample of observations, and we fit a complementary log-log model𝜋_𝑖1=𝑃𝑌^∗_𝑖=1∣𝑥_𝑖=exp−exp𝑥_𝑖𝜃,𝜋_𝑖2=𝑃𝑌^∗_𝑖=1∣𝑥_(−1,𝑖)=exp−exp𝑥_(−1,𝑖)𝜃, (16) where 𝑥_𝑖=(1,𝑥_𝑖1,…,𝑥_{𝑖,𝑝−1})^′is the 𝑃×1 vector of covariates for the 𝑖th observation, and 𝜃=(𝜃₀,…,𝜃_𝑝−1)^′ is the 𝑃×1 vector of unknown parameters. The dichotomized 𝜔^∗_𝑗 parameter corresponding to the effect 𝜃_𝑗 is𝜔^∗_𝑗=ln𝜋₁ln𝜋₂=ln𝑃𝑌^∗=1∣𝑥ln𝑃𝑌^∗=1∣𝑥_(−1,𝑗)=exp(𝑥𝜃)exp𝑥_(−1,𝑗)𝜃=exp𝜃_𝑗𝑗=0,…,𝑝−1. (17) In general, maximum likelihood estimation (MLE) can be used to estimate the parameter𝜃=(𝜃₀,…,𝜃_𝑝−1). Let 𝜃=(𝜃₀,…,𝜃_𝑝−1)^′ be the 𝑃×1 ML estimate of 𝜃, and let COV(𝜃) be the 𝑃×𝑃covariance matrix of 𝜃. Using COV(𝜃) from (23), one can construct confidence intervals. This matrix has as its diagonal the estimated variances of each of the ML estimates. The (𝑗+1)th diagonal element is given by 𝜎²𝜃_𝑗. Therefore,𝜔^∗_𝑗=exp𝜃_𝑗, (18) and for large samples, (𝜃^𝐿_𝑗,𝜃^𝑈_𝑗)=(𝜃_𝑗−𝑧_𝛼/2𝜎𝜃_𝑗,𝜃_𝑗+𝑧_𝛼/2𝜎𝜃_𝑗) is a 100(1−𝛼) percent confidence interval for the true 𝜃_𝑗. Then(exp(𝜃^𝐿_𝑗),exp(𝜃^𝑈_𝑗)) is a 100(1−𝛼) percent confidence interval for the true 𝜔^∗_𝑗.

We now compare the 𝜔_𝑗 from (7) with the 𝜔^∗_𝑗 from (17)𝜔_𝑗=ln𝜋₁ln𝜋₂𝜔^∗_𝑗=ln𝜋₁ln𝜋₂⟹𝜔^∗_𝑗=𝜔_𝑗𝜋⟹exp√6⋅𝛽_𝑗𝜎=exp𝜃_𝑗⟹𝜋√6⋅𝛽_𝑗𝜎=𝜃_𝑗∀𝛽_𝑗,𝜃_𝑗,𝜎. (19) This show that the coefficient of the complementary log-log model, 𝜃_𝑗, can be interpreted in terms of the regression coefficients, 𝛽_𝑗. Note that 𝛽 are related to the responses through the general linear regression model𝑦_𝑖=𝑥_𝑖𝛽+𝐸_𝑖𝑖=1,…,𝑛, (20) where the independent 𝐸_𝑖 are distributed as an extreme value with mean 0 and variance 𝜎²>0.

4. Covariance Matrix of Model Parameter Estimators

4.1. Derivation of var(𝜔^∗_𝑗) for Large 𝑛

The information matrix of generalized linear models has the form ∫=𝑋^′𝑊𝑋 [1], where 𝑊 is the diagonal matrix with diagonal elements 𝑤_𝑖=(𝜕𝜇_𝑖/𝜕𝜂_𝑖)²/(var(𝑦_𝑖)), 𝑦 is response variable with independent observations (𝑦₁,…𝑦_𝑛), and 𝑥_𝑖𝑗denote the value of predictor 𝑗,𝜇_𝑖=𝐸𝑦_𝑖,𝜂_𝑖=𝑔𝜇_𝑖=_𝑗𝜃_𝑗𝑥_𝑖𝑗,𝑗=0,1,…,𝑝−1. (21) The covariance matrix of 𝜃 is estimated by(𝑋^′𝑊𝑋)⁻¹.

Maximum likelihood estimation for the complementary log-log model is a special case of the generalized linear models. Let𝜇_𝑖=𝜋_𝑖=exp−exp_𝑗𝜃_𝑗𝑥_𝑖𝑗⟹𝜋_𝑖=exp−exp𝜂_𝑖,𝜕𝜇_𝑖𝜕𝜂_𝑖=−exp𝜂_𝑖^′exp−exp𝜂_𝑖=𝜋_𝑖ln𝜋_𝑖,𝑤_𝑖=𝜋_𝐼ln𝜋_𝑖²𝜋_𝑖1−𝜋_𝑖=𝜋_𝑖ln𝜋_𝑖²1−𝜋_𝑖, (22)

then𝑋^′𝑊𝑋=^𝑛𝑖=1𝜋_𝑖ln𝜋_𝑖²1−𝜋_𝑖^𝑛_𝑖=1𝑥_𝑖1𝜋_𝑖ln𝜋_𝑖²1−𝜋_𝑖⋯^𝑛_𝑖=1𝑥_{𝑖,𝑝−1}𝜋_𝑖ln𝜋_𝑖²1−𝜋_{𝑖𝑛𝑖=1}𝑥_𝑖1𝜋_𝑖ln𝜋_𝑖²1−𝜋_𝑖^𝑛_𝑖=1𝑥²_𝑖1𝜋_𝑖ln𝜋_𝑖²1−𝜋_𝑖⋯^𝑛_𝑖=1𝑥₁𝑥_{𝑖,𝑝−1}𝜋_𝑖ln𝜋_𝑖²1−𝜋_𝑖⋮⋮_𝑛𝑖=1𝑥_{𝑖,𝑝−1}𝜋_𝑖ln𝜋_𝑖²1−𝜋_𝑖^𝑛_𝑖=1𝑥_1𝑖𝑥_{𝑖,𝑝−1}𝜋_𝑖ln𝜋_𝑖²1−𝜋_𝑖⋯^𝑛_𝑖=1𝑥²_{𝑖,𝑝−1}𝜋_𝑖ln𝜋_𝑖²1−𝜋_𝑖. (23)It is readily shown that the results hold true for the largest extreme value distribution (Appendix A).

In large samples, var(𝜃_𝑗) approaches 𝜎²_𝜃𝑗|_𝜃=𝜃 [14] which equals the (𝑗+1)th diagonal element of (𝑋^′𝑊𝑋)⁻¹.

By applying the delta method, let 𝑓(𝜃_𝑗)=exp(𝜃_𝑗), then var𝜔^∗_𝑗⟶varexp𝜃_𝑗=var𝑓𝜃_𝑗=𝜕𝑓𝜃_𝑗𝜕𝜃_𝑗𝜃_{𝑗=𝜃𝑗}²var𝜃_𝑗=exp𝜃_𝑗²×𝜎²𝜃_𝑗. (24)

4.2. Derivation of var(𝜔_𝑗) for Large 𝑛

In large samples, from (10) 𝜎²→𝜎² [15]. Therefore,var𝜔_𝑗𝜋𝛽=varexp^𝑗𝜎√6𝜋𝛽⟶varexp^𝑗𝜎√6. (25) In addition, var(𝛽_𝑗)=𝜎²𝛿_𝑗.

By applying the delta method, let 𝑔(𝛽_𝑗)=exp(𝜋𝛽_𝑗/(𝜎√6)), thenvar𝜔_𝑗𝜋𝛽⟶varexp^𝑗𝜎√6=var𝑔𝛽_𝑗=𝜕𝑔𝛽_𝑗𝜕𝛽_𝑗𝛽_{𝑗=𝛽𝑗}²×var𝛽_𝑗=𝜋𝜎√6exp𝜋𝛽_𝑗𝜎√6²𝜎²𝛿_𝑗=𝜋²√6𝛿_𝑗exp𝜋𝛽_𝑗𝜎√6². (26)

5. Sample Sizes Saving

5.1. The Power for the Dichotomized Method

In large samples, 𝜎𝜃_𝑗 converges to 𝜎𝜃_𝑗 almost surely [14]. Therefore, for a given value of 𝜔_𝑗=exp𝜃_𝑗 (i.e., ln𝜔_𝑗=𝜃_𝑗), the power is given by𝑝𝜔_𝑗=𝑝rejectionof𝜔_𝑗=1∣𝜔_𝑗𝜃≠1=𝑝exp^𝐿_𝑗>1∣𝜃_𝑗+𝑝exp𝜃^𝑈_𝑗<1∣𝜃_𝑗=𝑝𝜃_𝑗>𝑧_𝛼/2𝜎𝜃_𝑗∣𝜃_𝑗+𝑝𝜃_𝑗<−𝑧_𝛼/2𝜎𝜃_𝑗∣𝜃_𝑗𝑧=𝑝𝑍>^𝛼/2𝜎𝜃_𝑗−ln𝜔_𝑗𝜎𝜃_𝑗+𝑝𝑍<−𝑧_𝛼/2𝜎𝜃_𝑗−ln𝜔_𝑗𝜎𝜃_𝑗=𝑝𝑍>𝑧_𝛼/2−ln𝜔_𝑗𝜎𝜃_𝑗+𝑝𝑍<−𝑧_𝛼/2−ln𝜔_𝑗𝜎𝜃_𝑗=𝑃𝑍>𝑧^•₁+𝑃𝑍<−𝑧^•₂, (27) where 𝑧^•₁=𝑧_𝛼/2−ln𝜔_𝑗𝜎𝜃_𝑗𝑧^•₂=𝑧_𝛼/2+ln𝜔_𝑗𝜎𝜃_𝑗. (28)

5.2. The Power for the Proposed Method

In large samples, 𝜎 converges to 𝜎 almost surely [15]. Therefore, for a given value of 𝜔_𝑗=exp(𝜋𝛽_𝑗/𝜎√6) (i.e., 𝛽_𝑗=𝜎(ln𝜔_𝑗√6/𝜋)), the power is given by 𝑝𝜔_𝑗𝜋=𝑝exp√6𝛽_𝑗𝜎^𝐿>1∣𝜔_𝑗𝜋+𝑝exp√6𝛽_𝑗𝜎^𝑈<1∣𝜔_𝑗=𝑃𝛽^𝐿_𝐽>𝑧_𝛼/2𝜎𝛿_𝑗∣𝛽_𝑗=𝜎ln𝜔_𝑗√6𝜋+𝑃𝛽^𝑈_𝐽<−𝑧_𝛼/2𝜎𝛿_𝑗∣𝛽_𝑗=𝜎ln𝜔_𝑗√6𝜋𝑧=𝑝𝑍>^𝛼/2𝜎𝛿_𝑗−𝜎ln𝜔_𝑗√6/𝜋𝜎𝛿_𝑗+𝑝𝑍<−𝑧_𝛼/2𝜎𝛿_𝑗−𝜎ln𝜔_𝑗√6/𝜋𝜎𝛿_𝑗=𝑝𝑍>𝑧_𝛼/2−ln𝜔_𝑗√6𝜋𝛿_𝑗+𝑝𝑍<−𝑧_𝛼/2−ln𝜔_𝑗√6𝜋𝛿_𝑗=𝑝𝑍>𝑧₁+𝑝𝑍<−𝑧₂, (29)

where 𝑧₁=𝑧_𝛼/2−ln𝜔_𝑗√6𝜋𝛿_𝑗𝑧₂=𝑧_𝛼/2+ln𝜔_𝑗√6𝜋𝛿_𝑗. (30) Our proposed method, since it is based on continuous data rather than dichotomized, is likely to be more powerful. We show that the proposed method can produce substantial sample size saving for a given power. Let(i)the number of parameters 𝑝=2 (i.e., 𝜃=(𝜃₀,𝜃₁)),(ii)𝑥_𝑖=(1,𝑥_𝑖1)^′, 𝑥_𝑖1∈{−𝑎+(2𝑎𝑛/(𝑔−1))∣𝑛=0,…,𝑔−1}, that is, 𝑥_𝑖1follows a discrete uniform distribution with range (−𝑎,𝑎). For simplicity, 𝑎=2.(iii)Total samples are 𝑛 and 𝑛^∗ for the proposed and dichotomized methods, respectively. These samples included 𝑘 and 𝑘^∗ set of these 𝑔 uniformly distributed points for the proposed and dichotomized methods, respectively. That is, 𝑛=𝑔𝑘 and 𝑛^∗=𝑔𝑘^∗, then𝛿_𝑗=𝑘^𝑔_𝑖=1𝑥_1𝑖−𝑥_1.²⁻¹,𝑗=1, (31) and from (23),𝜎²_𝜃𝑗=∑^𝑔_𝑖=1𝜋_𝑖ln𝜋_𝑖²/ln1−𝜋_𝑖(𝑘^∗)∑^𝑔_𝑖=1𝑥²_1𝑖𝜋_𝑖ln𝜋_𝑖²/ln1−𝜋_𝑖∑^𝑔_𝑖=1𝜋_𝑖ln𝜋_𝑖²/ln1−𝜋_𝑖−∑^𝑔_𝑖=1𝑥_1𝑖𝜋_𝑖ln𝜋_𝑖²/ln1−𝜋_𝑖². (32)We consider the same power for two methods:^{𝑧1=𝑧∗1}_{𝑧2=𝑧∗2}⟹^{𝑧𝛼/2−ln𝜔𝑗𝜎𝜃𝑗=𝑧𝛼/2−ln𝜔𝑗√6𝜋𝛿𝑗}_{𝑧𝛼/2+ln𝜔𝑗𝜎𝜃𝑗=𝑧𝛼/2+ln𝜔𝑗√6𝜋𝛿𝑗}⟹𝜋√6𝛿_𝑗=𝜎_𝜃𝑗𝜋,𝑗=1⟹√6𝑘^𝑔_𝑖=1𝑥_1𝑖−𝑥_1.²⁻¹=∑^𝑔_𝑖=1𝜋_𝑖ln𝜋_𝑖²/ln1−𝜋_𝑖(𝑘^∗)∑^𝑔_𝑖=1𝑥²_1𝑖𝜋_𝑖ln𝜋_𝑖²/ln1−𝜋_𝑖∑^𝑔_𝑖=1𝜋_𝑖ln𝜋_𝑖²/ln1−𝜋_𝑖−∑^𝑔_𝑖=1𝑥_1𝑖𝜋_𝑖ln𝜋_𝑖²/ln1−𝜋_𝑖² (33)relative sample size𝑛^∗𝑛=𝑘^∗𝑘=6𝜎²_𝜃𝑗𝜋²𝛿_𝑗=∑^𝑔_𝑖=1𝑥_1𝑖−𝑥_1.²×∑^𝑔_𝑖=1𝜋_𝑖ln𝜋_𝑖²/ln1−𝜋_𝑖𝜋²/6∑^𝑔_𝑖=1𝑥²_1𝑖𝜋_𝑖ln𝜋_𝑖²/ln1−𝜋_𝑖∑^𝑔_𝑖=1𝜋_𝑖ln𝜋_𝑖²/ln1−𝜋_𝑖−∑^𝑔_𝑖=1𝑥_1𝑖𝜋_𝑖ln𝜋_𝑖²/ln1−𝜋_𝑖². (34)That is, (34) is independent of 𝜎² and applies for any power, and any test size 𝛼.

Table 1 presents relative sample sizes 𝑛^∗/𝑛 for a given fixed parameter 𝜔^∗_𝑗and an average proportion of success 𝜋. We consider the situations in which 𝜋=∑^𝑔_𝑖=1(𝜋_𝑖/𝑔)=0.1,0.2,0.3,0.4,0.5, 𝑔=9, 𝜔^∗_𝑗=0.25,0.50,0.75.

Table 1 

Relative sample sizes required to attain any power for the dichotomizing method versus the proposed method.

For given fixed 𝜔^∗_𝑗 and 𝜋, the relative sample sizes in Table 1 can be computed by the following step:(i)compute the value 𝜃_𝑗 via the equation 𝜃_𝑗=ln(𝜔^∗_𝑗),(ii) calculate the cut-off point 𝐶 iteratively such that 𝜋 attained the specified value for the values 𝑥_𝑖1, using the value of 𝜃_𝑗in (i).

As can be seen from Table 1, all values are greater than 1. The values of 𝑛^∗/𝑛 increase as the𝜔^∗_𝑗 moves farther away from 1. Values of Table 1 immediately highlight the improvement accomplished by the proposed method.

6. Relative Efficiency of 𝜔_𝑗 with 𝜔^∗_𝑗

Here, we examine the relative efficiency of the estimate𝜔_𝑗 to the estimate 𝜔^∗_𝑗.

Using (24) and (26), the relative efficiency is given byr.e.𝜔_𝑗,𝜔^∗_𝑗=var𝜔^∗_𝑗var𝜔_𝑗=6exp𝜃_𝑗²×𝜎²𝜃_𝑗𝜋²𝛿_𝑗exp𝜆𝛽_𝑗/𝜎²=6𝜎²𝜃_𝑗𝜋²𝛿_𝑗. (35) Note that the relative efficiency is independent of 𝑛 and 𝜎² and converges to a constant. Comparing (34) and (35), the relative efficiency equals the relative sample sizes. Therefore, as in Table 1, the proposed method is a consistent improvement over the dichotomizing method with respect to relative efficiencies.

It should be noted that these results hold true under the following assumptions:(1)the responses 𝑦_𝑖 and 𝛽 are related through the equation 𝑦_𝑖=𝑥_𝑖𝛽+𝐸_𝑖 where the independent𝐸_𝑖 are distributed as an extreme value with mean 0 and variance 𝜎²>0,(2)the independent variables 𝑥_𝑖 follow a discrete uniform distribution.

7. Odds Ratio

For values of 𝜋 larger than 0.90, −ln(𝜋) and 𝜋/(1−𝜋)are very close. Hence, for large values of 𝜋,ln𝜋₁ln𝜋₂≅𝜋₁/1−𝜋₁𝜋₂/1−𝜋₂=OR. (36) And from (7), odds ratio is given by𝜋OR=exp√6⋅𝛽_𝑗𝜎. (37) The parameters estimated from the linear regression can be interpreted as an odds ratio.

8. Simulation Study

It should be noted that, as in Table 1, the proposed method is a consistent improvement over the dichotomizing method with respect to relative efficiencies. These results hold true under the assumption that predictor variable has a discrete uniform distribution and that the random variables 𝐸_𝑖 follow an extreme value distribution. To demonstrate the robustness of this conclusion to changes in the distributions of predictor variables, simulations were run under different distributional conditions. The data were sampled 10000 times for three sample sizes {𝑛=250,500,1000}, three average proportions of successes {𝜋=0.10,0.50,0.95}, and seven 𝜔_𝑗{𝜔_𝑗=0.75,0.90,1.1,1.2,1.3,1.4,1.5}. The simulated data are generated using the following algorithm(1)Generate 𝑦_𝑖, where 𝑦_𝑖=𝛽₀+𝛽₁𝑥_𝑖+𝐸_𝑖, 𝛽₁=√6ln𝜔_𝑗/𝜋 through (7) to produce the correct 𝜔_𝑗, and for simplicity 𝛽₀=0, 𝜎²=1.(2)For fixed 𝜋, generate cutoff point 𝐶 using (15).

We simulated the data for two scenarios based on the distribution of the explanatory variable. In the first scenario, the independent variable follows a continuous uniform distribution and range (−2, 2), and in the second, the independent variable follows a truncated normal distribution with mean 0 and range (−2, 2). The relative mean square errors, relative interval lengths, absolute biases, and the probability of coverage were calculated.

Results of the simulations addressing the validity of the proposed method are displayed in Tables 2 and 3.

Table 2 

Simulated relative mean square errors, relative intervals lengths, coverage probabilities, and absolute biases for the proposed and dichotomizing methods (using a continuous uniform distribution for the explanatory variable and an extreme value distribution for the errors).

Table 3 

Simulated relative mean square errors, relative intervals lengths, coverage probabilities, and absolute biases for the proposed and dichotomizing methods (using a truncated normal distribution for the explanatory variable and an extreme value distribution for the errors).

The simulations show that the relative mean square errors are all greater than 1, increasing with the average proportion of successes and when the 𝜔_𝑗 moves farther away from 1. The results in Tables 1 and 2 demonstrate that the proposed method provides confidence intervals which successfully maintain their nominal 95 percent coverage. For the proposed method in first scenario, 51 out of 63 coverage probabilities fell within (0.94, 0.96), and all 63 coverage probabilities are greater than 0.93 and, in the second scenario, almost all coverage probabilities fell within (0.94, 0.96). The absolute biases for proposed method are never greater than a few percent. The proposed method is less biased than the dichotomizing method in 6 of 63 simulations in both two scenarios.

9. An Example

To illustrate the application of the proposed method presented in the previous section, we utilize the data arising from the National Health Survey in Iran. The other analyses using this data appear in many places [16].

In this study, 14176 women aged 20–69 years were investigated. BMI (body mass index), our dependent variable, was calculated as weight in kilograms divided by height in meters squared (kg/m²). Independent variables included place of residence, age, smoking, economic index, marital status, and education level. The independent variables considered were both categorical and continuous. At first, BMI was treated as a continuous variable, and 𝜔_𝑗 and 95 percent confidence intervals were calculated using the proposed linear regression method. Then subjects were classified into obese (BMI ≥ 30 kg/m²) and nonobese (BMI <30 kg/m²). A complementary log-log model was used for the binary analysis, with obese or nonobese used as the outcome measure. The 𝜔^∗_𝑗 and 95 percent confidence intervals were calculated using the dichotomized method. Table 4 presents the coefficient estimates, estimated confidence intervals, and relative confidence interval lengths. The proposed and dichotomizing methods produced different confidence intervals, although the 𝜔_𝑗 and 𝜔^∗_𝑗 were similar only varying slightly. The 𝜔_𝑗estimate from the proposed method had smaller variances and shorter confidence intervals than the dichotomizing method. All relative confidence interval lengths were greater than 2.58.

Table 4 

Adjusted 𝜔^∗_𝑗, 𝜔_𝑗for obesity and confidence intervals using two methods for the National Health Survey.

10. Discussion

When assuming the errors𝐸_𝑖are distributed as an extreme value distribution, as noted before, the method has several advantages. First, the method allows the researcher to apply the complementary log-log model without dichotomizing and without loss of information. Second, the 𝜔^∗_𝑗 from the dichotomizing method is dependent on the chosen cutoff point 𝐶 and will vary with 𝑐. However, the proposed 𝜔_𝑗 is independent of the 𝑐 since 𝜔_𝑗 is a function of the continuous 𝑌_𝑖and not a function of the dichotomized 𝑌^∗_𝑖defined through 𝐶. Third, we show that the coefficient of the complementary log-log model, 𝜃_𝑗, can be interpreted in terms of the regression coefficients, 𝛽_𝑗. Fourth, when the independent variables 𝑥_𝑖 follow a discrete uniform distribution, the proposed method is a consistent improvement over the dichotomizing method with respect to relative efficiencies. The proposed method can provide sample size saving, smaller variances, and shorter confidence intervals than the dichotomized method. Fifth, when 𝜋is large, the parameters estimated from the linear regression can be interpreted as odds ratios.

Our results were consistent with the findings by Moser and Coombs [12] and Bakhshi et al. [16] showing the greater efficiency of parameter estimates from the regression method that avoids dichotomizing in comparison with a more traditional dichotomizing method using the logistic regression.

Our main recommendation is to let continuous response remain continuous. Do not throw away information by transforming the data to binary. This means that if the objective is to estimate and/or test coefficients when responses are continuous, please resist dichotomizing your response variable.

Appendix

A. Largest Extreme Value Distribution

(a) The PDF and CDF are given by 𝑓=𝜋𝑦∣𝑥𝛽,𝜎𝜎√6×exp−𝑦−𝑥𝛽−𝑘𝜎𝜎×𝜋√6−exp𝑦−𝑥𝛽−𝑘𝜎𝜎×𝜋√6⟨⟩𝑃−−∞⟨𝑥∞,𝜎0,𝑦≤𝑐=1−exp−exp𝑐−𝑥𝛽−𝑘𝜎𝜎×𝜋√6−∞⟨𝑥⟨∞,𝜎⟩0, (A.1) where 𝑌 is a continuous outcome variable, 𝑥=(1,𝑥₁,…,𝑥_𝑝−1) is the 𝑝×1 vector of known independent variables, 𝛽=(𝛽₀,𝛽₁,…,𝛽_𝑝−1) is the 𝑝×1 vector of unknown parameters, and 𝑘≈0.45.

It is easy to check that𝜔_𝑗=ln1−𝜋₁ln1−𝜋₂=ln1−𝑝𝑦≤𝑐∣𝑥ln1−𝑝𝑦≤𝑐∣𝑥_(−1,𝑗)=−exp−𝑐−𝑥^′𝛽−𝑘𝜎/𝜎×𝜋/√6−exp−𝑐−𝑥^′_−1,𝑗𝛽−𝑘𝜎/𝜎×𝜋/√6𝜋=exp√6⋅𝛽_𝑗𝜎⟹1−𝜋₁=1−𝜋₂^{√exp((𝜋/6)⋅(𝛽𝑗/𝜎))}, (A.2) where𝑥=1,𝑥₁,…,𝑥_𝑗,…,𝑥_𝑝−1,𝑥_(−1,𝑗)=1,𝑥₁,…,𝑥_𝑗−1,…,𝑥_𝑝−1,𝛽=𝛽₀,𝛽₁,…,𝛽_𝑗,…,𝛽_𝑝−1^′. (A.3)

 

(b) Suppose that 𝐸_𝑖 is distributed as a largest extreme value with mean 0 and variance 𝜎²>0. We conclude that the PDF and CDF of each independent 𝑌_𝑖 are given by (A.1), and the corresponding equality (A.2) is given by𝜔_𝑗=ln1−𝜋₁ln1−𝜋₂𝜋=exp√6⋅𝛽_𝑗𝜎. (B.1)

(c) Similar to largest extreme value distribution 𝜇_𝑖=𝜋_𝑖=1−exp−exp_𝑗𝜃_𝑗𝑥_𝑖𝑗⟹𝜋_𝑖=1−exp−exp𝜂_𝑖,𝜕𝜇_𝑖𝜕𝜂_𝑖=−−exp𝜂_𝑖^′exp−exp𝜂_𝑖=−1−𝜋_𝑖ln1−𝜋_𝑖𝑤_𝑖=1−𝜋_𝑖ln1−𝜋_𝑖²𝜋_𝑖1−𝜋_𝑖=1−𝜋_𝑖ln1−𝜋_𝑖²𝜋_𝑖, (C.1) then𝑋^′𝑊𝑋=^{𝑛𝑖=11−𝜋𝑖ln1−𝜋𝑖2𝜋𝑖𝑛𝑖=11−𝜋𝑖ln1−𝜋𝑖2𝜋𝑖⋯𝑛𝑖=1𝑥𝑖,𝑝−11−𝜋𝑖ln1−𝜋𝑖2𝜋𝑖𝑛𝑖=1𝑥𝑖11−𝜋𝑖ln1−𝜋𝑖2𝜋𝑖𝑛𝑖=1𝑥2𝑖11−𝜋𝑖ln1−𝜋𝑖2𝜋𝑖⋯𝑛𝑖=1𝑥1𝑥𝑖,𝑝−11−𝜋𝑖ln1−𝜋𝑖2𝜋𝑖}_{⋮⋮𝑛𝑖=1𝑥𝑖,𝑝−11−𝜋𝑖ln1−𝜋𝑖2𝜋𝑖𝑛𝑖=1𝑥𝑖𝑥𝑖,𝑝−11−𝜋𝑖ln1−𝜋𝑖2𝜋𝑖⋯𝑛𝑖=1𝑥2𝑖,𝑝−11−𝜋𝑖ln1−𝜋𝑖2𝜋𝑖}. (C.2)

Conflict of Interests

The authors have declared no conflict of interests.