# Let Continuous Outcome Variables Remain Continuous

Authors: Enayatollah Bakhshi; Brian McArdle; Kazem Mohammad; Behjat Seifi; Akbar Biglarian
Journal: Computational and Mathematical Methods in Medicine
 (2012)

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## 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.


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## Body

## 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 covariateX 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

Lety1,y2,…,yn be n independent observations on y, and let x1,x2,…,xp-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 as(1)yi=β0+β1xi1+β2xi2+⋯+βp-1xip-1+Eii=1,2,…,n,
or(2)yi=xiβ+Eii=1,2,…,n,
where(3)xi=(1,xi1,xi2,…,xip-1).
To complete the model, we make the following assumptions:(1)
E(Ei)=0 for i=1,2,…,n,(2)
var(Ei)=σ2 for i=1,2,…,n,(3)
the independentEi  follows an extreme value distribution for i=1,2,…,n.Writing the model for each of then observations, in matrix form, we have(4)[y1y2..yn]=[1x11x12…x1p-11x21x22…x2p-1..1xn1xn2…xnp-1][β0β1..βp-1]+[E1E2..En],
or(5)y=Xβ+E.
The preceding three assumptions on Ei and yican be expressed in terms of this model:(1)
E(E)=0,(2)
cov(E)=σ2I,(3)
theEi  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(6)f(y∣xβ,σ)=πσ6×exp⁡(-y-xβ-kσσ×π6-exp⁡(y-xβ-kσσ×π6))-∞〈x〈∞,σ〉0,P(y≤c)=exp⁡(-exp⁡(-c-xβ+kσσ×π6))-∞〈x〈∞,σ〉,k≈0.45.It is easy to check that(7)ωj=ln⁡π1ln⁡π2=ln⁡(p(y≤c∣x))ln⁡(p(y≤c∣x(-1,j)))=-exp⁡(-((c-x′β+kσ)/σ)×π/6)-exp⁡(-((c-x(-1,j)′β+kσ)/σ)×π/6)=exp⁡(π6⋅βjσ)⟹π1=π2exp⁡((π/6)⋅(βj/σ)),
where(8)x=(1,x1,…,xj,…,xp-1),x(-1,j)=(1,x1,…,xj-1,…,xp-1),β=(β0,β1,…,βj,…,βp-1)′.
To return to a random sample of observations (y1,y2,…,yn), we conclude that the PDF and CDF of each independent yi are given by (6), and the corresponding equality (7) is given by(9)ln⁡π̂1ln⁡π̂2=exp⁡(πσ̂6β̂j),
where the estimate β̂jis the (j+1)th element of vector β̂=(β̂0,β̂1,…,β̂j,…,β̂p-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

Let(10)β̂=(β̂0,β̂1,…,β̂j,…,β̂p-1)′=(X′X)-1X′Yj=0,…,p-1,σ̂2=Y′(In-X(X′X)-1X′)Y(n-p).
According to the preceding three assumptions on Ei andyi, we obtain(11)E(β̂)=E[(X′X)-1X′Y]=(X′X)-1X′EY=(X′X)-1X′Xβ=β,E(σ̂2)=1n-pE(Y′(In-X(X′X)-1X′)Y)=1n-p{tr⁡[(In-X(X′X)-1X′)σ2I]+E(Y′)[In-X(X′X)-1X′]E(Y)}=1n-p{σ2tr⁡[In-X(X′X)-1X′]+β′X′[In-X(X′X)-1X′]Xβ}=1n-p{σ2[n-tr⁡(X(X′X)-1X′)]+β′X′Xβ-β′X′X(X′X)-1X′Xβ}=1n-p{σ2[n-tr⁡(X(X′X)-1X′)]+β′X′Xβ-β′X′Xβ[n-tr⁡(X(X′X)-1X′)]}=1n-pσ2[n-tr⁡(IP)]=1n-pσ2(n-p)=σ2.
Therefore, β̂ andσ̂2are unbiased estimators of β andσ2.We have assumed thatEi is distributed as an extreme value, and we use the approximation of the extreme value distribution of the errors Ei by the normal distribution. For normally distributed observations, β̂j/(σ̂δj) follows a noncentral t distribution with n-p degree of freedom and noncentrality parameter -∞<βj/(σδj)<∞,(12)1-α=P{t1-(α/2)[n-p,βj(σδj)]<β̂j(σ̂δj)<tα/2[n-p,βj(σδj)]},
where tα/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 δj 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], then(13)1-α=P{βj/(σδj)-zα/2[1+(βj2/(σ2δj)-zα/22)/2(n-p)]1/21-(zα/22/2(n-p))<β̂j(σ̂δj)<βj/(σδj)+zα/2[1+((βj2/(σ2δj)-zα/22)/2(n-p))]1/21-(zα/22/2(n-p))},(βjσ)U={β̂jσ̂[1-zα/222(n-p)]+zα/2[δj(1+((β̂j2/σ̂2δj)-zα/222(n-p)))]1/2},(βjσ)L={β̂jσ̂[1-zα/222(n-p)]-zα/2[δj(1+((β̂j2/σ̂2δj)-zα/222(n-p)))]1/2},Thus, we obtain an approximate 100(1-α) percent confidence interval for ωj(14){exp⁡[π6(βjσ)L],exp⁡[π6(βjσ)U]}.


## 2.1. Model

Lety1,y2,…,yn be n independent observations on y, and let x1,x2,…,xp-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 as(1)yi=β0+β1xi1+β2xi2+⋯+βp-1xip-1+Eii=1,2,…,n,
or(2)yi=xiβ+Eii=1,2,…,n,
where(3)xi=(1,xi1,xi2,…,xip-1).
To complete the model, we make the following assumptions:(1)
E(Ei)=0 for i=1,2,…,n,(2)
var(Ei)=σ2 for i=1,2,…,n,(3)
the independentEi  follows an extreme value distribution for i=1,2,…,n.Writing the model for each of then observations, in matrix form, we have(4)[y1y2..yn]=[1x11x12…x1p-11x21x22…x2p-1..1xn1xn2…xnp-1][β0β1..βp-1]+[E1E2..En],
or(5)y=Xβ+E.
The preceding three assumptions on Ei and yican be expressed in terms of this model:(1)
E(E)=0,(2)
cov(E)=σ2I,(3)
theEi  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(6)f(y∣xβ,σ)=πσ6×exp⁡(-y-xβ-kσσ×π6-exp⁡(y-xβ-kσσ×π6))-∞〈x〈∞,σ〉0,P(y≤c)=exp⁡(-exp⁡(-c-xβ+kσσ×π6))-∞〈x〈∞,σ〉,k≈0.45.It is easy to check that(7)ωj=ln⁡π1ln⁡π2=ln⁡(p(y≤c∣x))ln⁡(p(y≤c∣x(-1,j)))=-exp⁡(-((c-x′β+kσ)/σ)×π/6)-exp⁡(-((c-x(-1,j)′β+kσ)/σ)×π/6)=exp⁡(π6⋅βjσ)⟹π1=π2exp⁡((π/6)⋅(βj/σ)),
where(8)x=(1,x1,…,xj,…,xp-1),x(-1,j)=(1,x1,…,xj-1,…,xp-1),β=(β0,β1,…,βj,…,βp-1)′.
To return to a random sample of observations (y1,y2,…,yn), we conclude that the PDF and CDF of each independent yi are given by (6), and the corresponding equality (7) is given by(9)ln⁡π̂1ln⁡π̂2=exp⁡(πσ̂6β̂j),
where the estimate β̂jis the (j+1)th element of vector β̂=(β̂0,β̂1,…,β̂j,…,β̂p-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

Let(10)β̂=(β̂0,β̂1,…,β̂j,…,β̂p-1)′=(X′X)-1X′Yj=0,…,p-1,σ̂2=Y′(In-X(X′X)-1X′)Y(n-p).
According to the preceding three assumptions on Ei andyi, we obtain(11)E(β̂)=E[(X′X)-1X′Y]=(X′X)-1X′EY=(X′X)-1X′Xβ=β,E(σ̂2)=1n-pE(Y′(In-X(X′X)-1X′)Y)=1n-p{tr⁡[(In-X(X′X)-1X′)σ2I]+E(Y′)[In-X(X′X)-1X′]E(Y)}=1n-p{σ2tr⁡[In-X(X′X)-1X′]+β′X′[In-X(X′X)-1X′]Xβ}=1n-p{σ2[n-tr⁡(X(X′X)-1X′)]+β′X′Xβ-β′X′X(X′X)-1X′Xβ}=1n-p{σ2[n-tr⁡(X(X′X)-1X′)]+β′X′Xβ-β′X′Xβ[n-tr⁡(X(X′X)-1X′)]}=1n-pσ2[n-tr⁡(IP)]=1n-pσ2(n-p)=σ2.
Therefore, β̂ andσ̂2are unbiased estimators of β andσ2.We have assumed thatEi is distributed as an extreme value, and we use the approximation of the extreme value distribution of the errors Ei by the normal distribution. For normally distributed observations, β̂j/(σ̂δj) follows a noncentral t distribution with n-p degree of freedom and noncentrality parameter -∞<βj/(σδj)<∞,(12)1-α=P{t1-(α/2)[n-p,βj(σδj)]<β̂j(σ̂δj)<tα/2[n-p,βj(σδj)]},
where tα/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 δj 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], then(13)1-α=P{βj/(σδj)-zα/2[1+(βj2/(σ2δj)-zα/22)/2(n-p)]1/21-(zα/22/2(n-p))<β̂j(σ̂δj)<βj/(σδj)+zα/2[1+((βj2/(σ2δj)-zα/22)/2(n-p))]1/21-(zα/22/2(n-p))},(βjσ)U={β̂jσ̂[1-zα/222(n-p)]+zα/2[δj(1+((β̂j2/σ̂2δj)-zα/222(n-p)))]1/2},(βjσ)L={β̂jσ̂[1-zα/222(n-p)]-zα/2[δj(1+((β̂j2/σ̂2δj)-zα/222(n-p)))]1/2},Thus, we obtain an approximate 100(1-α) percent confidence interval for ωj(14){exp⁡[π6(βjσ)L],exp⁡[π6(βjσ)U]}.


## 3. Comparison of the Two Methods

LetYi be a continuous outcome variable. For fixed value of C, we define Yi* such that(15)Yi*={1ifYi≥C,0ifYi<C.
Suppose that Y1*,…,Yn* form a random sample of observations, and we fit a complementary log-log model(16)πi1=P(Yi*=1∣xi)=exp⁡(-exp⁡(xiθ)),πi2=P(Yi*=1∣x(-1,i))=exp⁡(-exp⁡(x(-1,i)θ)),
where xi=(1,xi1,…,xi,p-1)′is the P×1 vector of covariates for the ith observation, and θ=(θ0,…,θp-1)′ is the P×1 vector of unknown parameters. The dichotomized ωj* parameter corresponding to the effect θj is(17)ωj*=ln⁡(π1)ln⁡(π2)=ln⁡(P(Y*=1∣x))ln⁡(P(Y*=1∣x(-1,j)))=(exp⁡(xθ))(exp⁡(x(-1,j)θ))=exp⁡(θj)j=0,…,p-1.
In general, maximum likelihood estimation (MLE) can be used to estimate the parameterθ=(θ0,…,θp-1). Let θ̂=(θ̂0,…,θ̂p-1)′ be the P×1 ML estimate of θ, and let COV(θ̂) be the P×Pcovariance matrix of θ̂. Using COV(θ̂) 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 σθ̂j2. Therefore,(18)ω̂j*=exp⁡(θ̂j),
and for large samples, (θ̂jL,θ̂jU)=(θ̂j-zα/2σ̂θ̂j,θ̂j+zα/2σ̂θ̂j) is a 100(1-α) percent confidence interval for the true θj. Then(exp(θ̂jL),exp(θ̂jU)) is a 100(1-α) percent confidence interval for the true ωj*.We now compare theωj from (7) with the ωj* from (17)(19)ωj=ln⁡(π1)ln⁡(π2)ωj*=ln⁡(π1)ln⁡(π2)⟹ωj*=ωj⟹exp⁡(π6⋅βjσ)=exp⁡(θj)⟹π6⋅βjσ=θj∀βj,θj,σ.
This show that the coefficient of the complementary log-log model, θj, can be interpreted in terms of the regression coefficients, βj. Note that β are related to the responses through the general linear regression model(20)yi=xiβ+Eii=1,…,n,
where the independent Ei are distributed as an extreme value with mean 0 and variance σ2>0.


## 4. Covariance Matrix of Model Parameter Estimators

### 4.1. Derivation ofvar(ωj*) 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 wi=(∂μi/∂ηi)2/(var(yi)), y is response variable with independent observations (y1,…yn), and xijdenote the value of predictor j,(21)μi=E(yi),ηi=g(μi)=∑jθjxij,j=0,1,…,p-1.
The covariance matrix of θ̂ is estimated by(X′ŴX)-1.Maximum likelihood estimation for the complementary log-log model is a special case of the generalized linear models. Let(22)μi=πi=exp⁡(-exp⁡(∑jθjxij))⟹πi=exp⁡(-exp⁡(ηi)),∂μi∂ηi=(-exp⁡(ηi))′exp⁡(-exp⁡(ηi))=πiln⁡πi,wi=(πIln⁡πi)2πi(1-πi)=πi(ln⁡πi)21-πi,then(23)X′WX=[∑i=1nπi(ln⁡πi)21-πi∑i=1nxi1πi(ln⁡πi)21-πi⋯∑i=1nxi,p-1πi(ln⁡πi)21-πi∑i=1nxi1πi(ln⁡πi)21-πi∑i=1nxi12πi(ln⁡πi)21-πi⋯∑i=1nx1xi,p-1πi(ln⁡πi)21-πi⋮⋮∑i=1nxi,p-1πi(ln⁡πi)21-πi∑i=1nx1ixi,p-1πi(ln⁡πi)21-πi⋯∑i=1nxi,p-12πi(ln⁡πi)21-πi].It is readily shown that the results hold true for the largest extreme value distribution (Appendix A).In large samples,var(θ̂j) approaches σθj2|θ=θ̂ [14] which equals the (j+1)th diagonal element of (X′WX)-1.By applying the delta method, letf(θ̂j)=exp(θ̂j), then
(24)var(ω̂j*)⟶var(exp⁡(θ̂j))=var(f(θ̂j))=(∂f(θ̂j)∂θ̂j|θ̂j=θj)2(var(θ̂j))=(exp⁡(θj))2×σθ̂j2.


### 4.2. Derivation ofvar(ω̂j) for Large n

In large samples, from (10) σ̂2→σ2 [15]. Therefore,(25)var(ω̂j)=var(exp⁡(πβ̂jσ̂6))⟶var(exp⁡(πβ̂jσ6)).
In addition, var(β̂j)=σ2δj.By applying the delta method, letg(β̂j)=exp(πβ̂j/(σ6)), then(26)var(ω̂j)⟶var(exp⁡(πβ̂jσ6))=var(g(β̂j))=(∂g(β̂j)∂β̂j|β̂j=βj)2×var(β̂j)=(πσ6exp⁡(πβjσ6))2σ2δj=π26δj(exp⁡πβjσ6)2.


## 4.1. Derivation ofvar(ωj*) 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 wi=(∂μi/∂ηi)2/(var(yi)), y is response variable with independent observations (y1,…yn), and xijdenote the value of predictor j,(21)μi=E(yi),ηi=g(μi)=∑jθjxij,j=0,1,…,p-1.
The covariance matrix of θ̂ is estimated by(X′ŴX)-1.Maximum likelihood estimation for the complementary log-log model is a special case of the generalized linear models. Let(22)μi=πi=exp⁡(-exp⁡(∑jθjxij))⟹πi=exp⁡(-exp⁡(ηi)),∂μi∂ηi=(-exp⁡(ηi))′exp⁡(-exp⁡(ηi))=πiln⁡πi,wi=(πIln⁡πi)2πi(1-πi)=πi(ln⁡πi)21-πi,then(23)X′WX=[∑i=1nπi(ln⁡πi)21-πi∑i=1nxi1πi(ln⁡πi)21-πi⋯∑i=1nxi,p-1πi(ln⁡πi)21-πi∑i=1nxi1πi(ln⁡πi)21-πi∑i=1nxi12πi(ln⁡πi)21-πi⋯∑i=1nx1xi,p-1πi(ln⁡πi)21-πi⋮⋮∑i=1nxi,p-1πi(ln⁡πi)21-πi∑i=1nx1ixi,p-1πi(ln⁡πi)21-πi⋯∑i=1nxi,p-12πi(ln⁡πi)21-πi].It is readily shown that the results hold true for the largest extreme value distribution (Appendix A).In large samples,var(θ̂j) approaches σθj2|θ=θ̂ [14] which equals the (j+1)th diagonal element of (X′WX)-1.By applying the delta method, letf(θ̂j)=exp(θ̂j), then
(24)var(ω̂j*)⟶var(exp⁡(θ̂j))=var(f(θ̂j))=(∂f(θ̂j)∂θ̂j|θ̂j=θj)2(var(θ̂j))=(exp⁡(θj))2×σθ̂j2.


## 4.2. Derivation ofvar(ω̂j) for Large n

In large samples, from (10) σ̂2→σ2 [15]. Therefore,(25)var(ω̂j)=var(exp⁡(πβ̂jσ̂6))⟶var(exp⁡(πβ̂jσ6)).
In addition, var(β̂j)=σ2δj.By applying the delta method, letg(β̂j)=exp(πβ̂j/(σ6)), then(26)var(ω̂j)⟶var(exp⁡(πβ̂jσ6))=var(g(β̂j))=(∂g(β̂j)∂β̂j|β̂j=βj)2×var(β̂j)=(πσ6exp⁡(πβjσ6))2σ2δj=π26δj(exp⁡πβjσ6)2.


## 5. Sample Sizes Saving

### 5.1. The Power for the Dichotomized Method

In large samples,σ̂θ̂j converges to σθ̂j almost surely [14]. Therefore, for a given value of ωj=expθj (i.e., lnωj=θj), the power is given by(27)p(ωj)=p{rejectionofωj=1∣ωj≠1}=p{exp⁡(θjL)>1∣θj}+p{exp⁡(θjU)<1∣θj}=p{θ̂j>zα/2σθ̂j∣θj}+p{θ̂j<-zα/2σθ̂j∣θj}=p{Z>zα/2σθ̂j-ln⁡ωjσθ̂j}+p{Z<-zα/2σθ̂j-ln⁡ωjσθ̂j}=p{Z>zα/2-ln⁡ωjσθ̂j}+p{Z<-zα/2-ln⁡ωjσθ̂j}=P{Z>z1•}+P{Z<-z2•},
where
(28){z1•=zα/2-ln⁡ωjσθ̂jz2•=zα/2+ln⁡ωjσθ̂j}.


### 5.2. The Power for the Proposed Method

In large samples,σ̂ converges to σ almost surely [15]. Therefore, for a given value of ωj=exp(πβj/σ6) (i.e., βj=σ(lnωj6/π)), the power is given by
(29)p(ωj)=p{exp⁡(π6(βjσ)L)>1∣ωj}+p{exp⁡(π6(βjσ)U)<1∣ωj}=P{βJL>zα/2σδj∣βj=σln⁡ωj6π}+P{βJU<-zα/2σδj∣βj=σln⁡ωj6π}=p{Z>zα/2σδj-(σln⁡ωj6/π)σδj}+p{Z<-zα/2σδj-(σln⁡ωj6/π)σδj}=p{Z>zα/2-ln⁡ωj6πδj}+p{Z<-zα/2-ln⁡ωj6πδj}=p{Z>z1}+p{Z<-z2},where(30){z1=zα/2-ln⁡ωj6πδjz2=zα/2+ln⁡ωj6πδj}.
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 parametersp=2 (i.e., θ=(θ0,θ1)),(ii)
xi=(1,xi1)′, xi1∈{-a+(2an/(g-1))∣n=0,…,g-1}, that is, xi1follows a discrete uniform distribution with range (-a,a). For simplicity, a=2.(iii)
Total samples aren 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*, then(31)δj=[k∑i=1g(x1i-x̅1.)2]-1,j=1,
and from (23),(32)σθ̂j2=∑i=1g((πi)(ln⁡(πi))2/ln⁡(1-πi))(k*){∑i=1gx1i2((πi)(ln⁡(πi))2/ln⁡(1-πi))∑i=1g((πi)(ln⁡(πi))2/ln⁡(1-πi))-[∑i=1gx1i((πi)(ln⁡(πi))2/ln⁡(1-πi))]2}.We consider the same power for two methods:(33)z1=z1*z2=z2*⟹{zα/2-ln⁡ωjσθ̂j=zα/2-ln⁡ωj6πδjzα/2+ln⁡ωjσθ̂j=zα/2+ln⁡ωj6πδj⟹π6δj=σθ̂j,j=1⟹π6[k∑i=1g(x1i-x̅1.)2]-1=∑i=1g((πi)(ln⁡(πi))2/ln⁡(1-πi))(k*){∑i=1gx1i2((πi)(ln⁡(πi))2/ln⁡(1-πi))∑i=1g((πi)(ln⁡(πi))2/ln⁡(1-πi))-[∑i=1gx1i((πi)(ln⁡(πi))2/ln⁡(1-πi))]2}relative sample size(34)n*n=k*k=6σθ̂j2π2δj=∑i=1g(x1i-x̅1.)2×∑i=1g((πi)(ln⁡(πi))2/ln⁡(1-πi))(π2/6){∑i=1gx1i2((πi)(ln⁡(πi))2/ln⁡(1-πi))∑i=1g((πi)(ln⁡(πi))2/ln⁡(1-πi))-[∑i=1gx1i((πi)(ln⁡(πi))2/ln⁡(1-πi))]2}.That is, (34) is independent of σ2 and applies for any power, and any test size α.Table1 presents relative sample sizes n*/n for a given fixed parameter ωj*and an average proportion of success π̅. We consider the situations in which π̅=∑i=1g(πi/g)=0.1,0.2,0.3,0.4,0.5, g=9, ωj*=0.25,0.50,0.75.Table 1
Relative sample sizes required to attain any power for the dichotomizing method versus the proposed method.
ω*=exp(θ)Average proportion of successes (π̅)0.10.20.30.40.50.2523.71669.50927.49547.19966.85750.5010.67195.41763.42152.52092.17840.757.70883.87132.51711.93801.5841For given fixedωj* and π̅, the relative sample sizes in Table 1 can be computed by the following step:(i)
compute the valueθj via the equation θj=ln(ωj*),(ii)
calculate the cut-off pointC iteratively such that π̅ attained the specified value for the values xi1, using the value of θjin (i).As can be seen from Table1, all values are greater than 1. The values of n*/n increase as theωj* moves farther away from 1. Values of Table 1 immediately highlight the improvement accomplished by the proposed method.


## 5.1. The Power for the Dichotomized Method

In large samples,σ̂θ̂j converges to σθ̂j almost surely [14]. Therefore, for a given value of ωj=expθj (i.e., lnωj=θj), the power is given by(27)p(ωj)=p{rejectionofωj=1∣ωj≠1}=p{exp⁡(θjL)>1∣θj}+p{exp⁡(θjU)<1∣θj}=p{θ̂j>zα/2σθ̂j∣θj}+p{θ̂j<-zα/2σθ̂j∣θj}=p{Z>zα/2σθ̂j-ln⁡ωjσθ̂j}+p{Z<-zα/2σθ̂j-ln⁡ωjσθ̂j}=p{Z>zα/2-ln⁡ωjσθ̂j}+p{Z<-zα/2-ln⁡ωjσθ̂j}=P{Z>z1•}+P{Z<-z2•},
where
(28){z1•=zα/2-ln⁡ωjσθ̂jz2•=zα/2+ln⁡ωjσθ̂j}.


## 5.2. The Power for the Proposed Method

In large samples,σ̂ converges to σ almost surely [15]. Therefore, for a given value of ωj=exp(πβj/σ6) (i.e., βj=σ(lnωj6/π)), the power is given by
(29)p(ωj)=p{exp⁡(π6(βjσ)L)>1∣ωj}+p{exp⁡(π6(βjσ)U)<1∣ωj}=P{βJL>zα/2σδj∣βj=σln⁡ωj6π}+P{βJU<-zα/2σδj∣βj=σln⁡ωj6π}=p{Z>zα/2σδj-(σln⁡ωj6/π)σδj}+p{Z<-zα/2σδj-(σln⁡ωj6/π)σδj}=p{Z>zα/2-ln⁡ωj6πδj}+p{Z<-zα/2-ln⁡ωj6πδj}=p{Z>z1}+p{Z<-z2},where(30){z1=zα/2-ln⁡ωj6πδjz2=zα/2+ln⁡ωj6πδj}.
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 parametersp=2 (i.e., θ=(θ0,θ1)),(ii)
xi=(1,xi1)′, xi1∈{-a+(2an/(g-1))∣n=0,…,g-1}, that is, xi1follows a discrete uniform distribution with range (-a,a). For simplicity, a=2.(iii)
Total samples aren 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*, then(31)δj=[k∑i=1g(x1i-x̅1.)2]-1,j=1,
and from (23),(32)σθ̂j2=∑i=1g((πi)(ln⁡(πi))2/ln⁡(1-πi))(k*){∑i=1gx1i2((πi)(ln⁡(πi))2/ln⁡(1-πi))∑i=1g((πi)(ln⁡(πi))2/ln⁡(1-πi))-[∑i=1gx1i((πi)(ln⁡(πi))2/ln⁡(1-πi))]2}.We consider the same power for two methods:(33)z1=z1*z2=z2*⟹{zα/2-ln⁡ωjσθ̂j=zα/2-ln⁡ωj6πδjzα/2+ln⁡ωjσθ̂j=zα/2+ln⁡ωj6πδj⟹π6δj=σθ̂j,j=1⟹π6[k∑i=1g(x1i-x̅1.)2]-1=∑i=1g((πi)(ln⁡(πi))2/ln⁡(1-πi))(k*){∑i=1gx1i2((πi)(ln⁡(πi))2/ln⁡(1-πi))∑i=1g((πi)(ln⁡(πi))2/ln⁡(1-πi))-[∑i=1gx1i((πi)(ln⁡(πi))2/ln⁡(1-πi))]2}relative sample size(34)n*n=k*k=6σθ̂j2π2δj=∑i=1g(x1i-x̅1.)2×∑i=1g((πi)(ln⁡(πi))2/ln⁡(1-πi))(π2/6){∑i=1gx1i2((πi)(ln⁡(πi))2/ln⁡(1-πi))∑i=1g((πi)(ln⁡(πi))2/ln⁡(1-πi))-[∑i=1gx1i((πi)(ln⁡(πi))2/ln⁡(1-πi))]2}.That is, (34) is independent of σ2 and applies for any power, and any test size α.Table1 presents relative sample sizes n*/n for a given fixed parameter ωj*and an average proportion of success π̅. We consider the situations in which π̅=∑i=1g(πi/g)=0.1,0.2,0.3,0.4,0.5, g=9, ωj*=0.25,0.50,0.75.Table 1
Relative sample sizes required to attain any power for the dichotomizing method versus the proposed method.
ω*=exp(θ)Average proportion of successes (π̅)0.10.20.30.40.50.2523.71669.50927.49547.19966.85750.5010.67195.41763.42152.52092.17840.757.70883.87132.51711.93801.5841For given fixedωj* and π̅, the relative sample sizes in Table 1 can be computed by the following step:(i)
compute the valueθj via the equation θj=ln(ωj*),(ii)
calculate the cut-off pointC iteratively such that π̅ attained the specified value for the values xi1, using the value of θjin (i).As can be seen from Table1, all values are greater than 1. The values of n*/n increase as theωj* moves farther away from 1. Values of Table 1 immediately highlight the improvement accomplished by the proposed method.


## 6. Relative Efficiency ofω̂j with ω̂j*

Here, we examine the relative efficiency of the estimateω̂j to the estimate ω̂j*.Using (24) and (26), the relative efficiency is given by(35)r.e.(ω̂j,ω̂j*)=var(ω̂j*)var(ω̂j)=6(exp⁡(θj))2×σθ̂j2π2δj(exp⁡(λβj/σ))2=6σθ̂j2π2δj.
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 responsesyi and β are related through the equation yi=xiβ+Ei where the independentEi are distributed as an extreme value with mean 0 and variance σ2>0,(2)
the independent variablesxi follow a discrete uniform distribution.


## 7. Odds Ratio

For values ofπ larger than 0.90, -ln(π) and π/(1-π)are very close. Hence, for large values of π,(36)ln⁡(π1)ln⁡(π2)≅π1/1-π1π2/1-π2=OR.
And from (7), odds ratio is given by(37)OR=exp⁡(π6⋅βjσ).
The parameters estimated from the linear regression can be interpreted as an odds ratio.


## 8. Simulation Study

It should be noted that, as in Table1, 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 Ei 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 {π̅=0.10,0.50,0.95}, and seven ωj{ωj=0.75,0.90,1.1,1.2,1.3,1.4,1.5}. The simulated data are generated using the following algorithm(1)
Generateyi, where yi=β0+β1xi+Ei, β1=6lnωj/π through (7) to produce the correct ωj, and for simplicity β0=0, σ2=1.(2)
For fixedπ̅, 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 Tables2 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).
Sample sizeω Cut off.75.91.11.21.31.41.51.15a1.071.091.141.241.471.711.10b1.031.031.071.141.231.350.100.943c0.9480.9490.9490.9450.9380.9330.948d0.9470.9490.9470.9510.9470.9530.05e0.040.120.140.100.150.110.07f0.010.170.130.240.340.581.231.261.271.281.271.241.262.161.131.231.141.151.171.1910000.500.9400.9510.9510.9450.9420.9370.9340.9510.9490.9510.9500.9480.9470.9480.040.010.080.100.050.090.040.050.040.150.120.090.120.1312.7512.4413.2212.6813.1412.9112.793.673.573.583.633.693.763.840.950.9430.9510.9520.9440.9440.9380.9290.9520.9540.9520.9520.9510.9510.9510.040.070.110.100.100.170.100.750.680.861.011.211.451.241.301.081.071.171.241.541.951.161.031.041.081.151.251.390.100.9420.9500.9510.950.9440.9410.9360.9510.9500.9490.9510.9540.9540.9530.120.070.240.250.210.180.290.230.080.330.390.410.731.211.351.101.271.261.261.251.261.261.031.131.141.161.171.205000.500.9400.9490.9470.9480.9430.9400.9330.9520.9510.9490.9490.9540.9500.9510.230.340.270.230.260.250.380.480.110.170.180.310.260.4213.0413.1713.813.9014.4514.4814.473.723.653.683.733.823.913.990.950.9420.9470.9510.9490.9470.9380.9350.9530.9520.9540.9550.9550.9530.9540.050.110.080.080.240.320.270.941.381.781.922.523.002.9013.4114.461.121.281.521.962.333.783.731.041.091.181.301.450.100.9420.9490.9490.9450.9420.9420.9330.9570.9540.9480.9490.9520.9570.9530.020.200.380.330.420.410.662.112.740.420.841.181.782.241.271.251.321.281.301.301.291.161.131.131.141.161.181.202500.500.9410.9480.9520.9470.9450.9430.9330.9510.9510.9510.9500.9510.9510.9510.120.130.350.440.410.530.550.110.220.390.470.510.740.5912.9814.615.6415.4617.0516.8918.333.753.723.823.884.014.124.290.950.9450.9550.9460.9480.9400.9370.9320.9590.9550.9550.9590.9580.9570.9520.020.160.390.220.460.470.511.222.753.973.984.995.196.19a: Relative mean square errors, b: Relative intervals lengths, c: Coverage probability (proposed), d: Coverage probability (dichotomized), e: % bias (proposed), f: % bias (dichotomized).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).
Sample sizeω Cut off.75.91.11.21.31.41.51.17a1.021.081.131.191.281.361.11b1.031.031.061.101.251.220.100.942c0.9480.9480.9520.9440.9420.9400.951d0.9510.9500.9520.9490.9510.9510.08e0.060.030.140.130.140.160.10f0.110.150.230.300.390.391.261.241.261.281.281.251.281.241.131.131.141.141.151.1710000.500.9440.9480.9520.9470.9470.9440.9410.9480.9510.9490.9490.9470.9500.9490.020.090.080.070.180.160.130.030.060.120.160.200.160.1412.3313.1213.0312.7112.8612.5512.883.623.593.613.623.643.683.710.950.9440.9510.9480.9480.9450.9450.9460.9520.9480.950.9490.9490.9510.9520.100.040.110.040.160.160.201.261.051.561.361.431.801.941.181.091.061.751.231.321.581.111.031.031.061.111.161.230.100.9450.950.9510.9510.9490.9430.9440.9530.9530.9530.9500.9490.9510.9500.040.130.310.180.330.360.370.210.080.370.500.620.690.961.251.271.271.291.271.291.251.141.131.131.141.151.161.175000.500.9440.9480.9490.9470.9480.9440.9350.9510.9510.9510.9480.9510.9480.9490.130.220.350.370.350.300.440.160.190.390.480.440.410.5413.1114.0214.0213.513.5413.8014.323.733.713.733.753.773.813.860.950.9440.950.9510.9500.9470.9440.9440.9540.950.9510.9530.9480.9560.9530.150.100.240.380.320.330.432.502.702.923.102.923.363.891.281.111.121.191.331.541.761.111.031.041.081.131.191.280.100.9470.9510.9500.9470.9500.9500.9420.9510.9500.9500.9520.9540.9520.9510.400.340.370.640.690.580.810.260.060.691.081.301.552.221.321.301.271.331.311.331.311.151.131.131.141.181.171.182500.500.9510.950.9530.9510.9400.9450.9400.9490.9510.9520.9480.9480.9500.9480.220.430.570.690.660.580.660.380.530.640.890.910.820.9114.0914.5116.2715.9115.8915.7315.603.863.873.933.923.984.044.110.950.9430.950.9510.9510.9470.9440.9370.9530.950.9530.9560.9530.9560.9520.300.370.570.680.420.620.754.985.526.5475.916.176.887.72a: Relative mean square errors, b: Relative intervals lengths, c: Coverage probability (proposed), d: Coverage probability (dichotomized), e: % bias (proposed), f: % bias (dichotomized).The simulations show that the relative mean square errors are all greater than 1, increasing with the average proportion of successes and when theωj 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 ω̂j 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 ω̂j* 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 ω̂j and ω̂j* were similar only varying slightly. The ω̂jestimate 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
Adjustedω̂j*, ω̂jfor obesity and confidence intervals using two methods for the National Health Survey.
Covariatesω̂j(ω̂j*)95% CIa (proposed)95% CI  (dichotomized)Relativeb length of CIPlace of residence1.65 (1.97)c1.58–1.741.79–2.182.43Age1.021 (1.019)1.018–1.0221.015–1.0221.75Years of education0.99 (0.98)0.985–0.9970.971–0.9941.92Smoking0.76 (0.68)0.66–0.900.51–0.921.71Marital status1.16 (1.42)1.10–1.221.27–1.582.58Lower-middle economy index1.24 (1.32)1.14–1.321.18–1.481.67Upper-middle economy index1.21 (1.26)1.14–1.291.12–1.422.0High economy index1.20 (1.21)1.11–1.301.08–1.361.47aconfidence interval, bdichotomized/proposed, cproposed (dichotomized).


## 10. Discussion

When assuming the errorsEiare 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 ω̂j* from the dichotomizing method is dependent on the chosen cutoff point C and will vary with c. However, the proposed ω̂j is independent of the c since ω̂j is a function of the continuous Yiand not a function of the dichotomized Yi*defined through C. Third, we show that the coefficient of the complementary log-log model, θj, can be interpreted in terms of the regression coefficients, βj. Fourth, when the independent variables xi 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.



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*Source: 639124-2012-05-29.xml*