Read the article and write a 2 page summary

Statistica Sinica 16(2006), 847-860

PSEUDO-R 2

IN LOGISTIC REGRESSION MODEL

Bo Hu, Jun Shao and Mari Palta

University of Wisconsin-Madison

Abstract: Logistic regression with binary and multinomial outcomes is commonly

used, and researchers have long searched for an interpretable measure of the strength

of a particular logistic model. This article describes the large sample properties

of some pseudo-R2 statistics for assessing the predictive strength of the logistic

regression model. We present theoretical results regarding the convergence and

asymptotic normality of pseudo-R2s. Simulation results and an example are also

presented. The behavior of the pseudo-R2s is investigated numerically across a

range of conditions to aid in practical interpretation.

Key words and phrases: Entropy, logistic regression, pseudo-R2

1. Introduction

Logistic regression for binary and multinomial outcomes is commonly used

in health research. Researchers often desire a statistic ranging from zero to one

to summarize the overall strength of a given model, with zero indicating a model

with no predictive value and one indicating a perfect fit. The coefficient of deter-

mination R2 for the linear regression model serves as a standard for such measures

(Draper and Smith (1998)). Statisticians have searched for a corresponding in-

dicator for models with binary/multinomial outcome. Many different R2 statis-

tics have been proposed in the past three decades (see, e.g., McFadden (1973),

McKelvey and Zavoina (1975), Maddala (1983), Agresti (1986), Nagelkerke

(1991), Cox and Wermuch (1992), Ash and Shwartz (1999), Zheng and Agresti

(2000)). These statistics, which are usually identical to the standard R2 when

applied to a linear model, generally fall into categories of entropy-based and

variance-based (Mittlböck and Schemper (1996)). Entropy-based R2 statistics,

also called pseudo-R2s, have gained some popularity in the social sciences (Mad-

dala (1983), Laitila (1993) and Long (1997)). McKelvey and Zavoina (1975)

proposed a pseudo-R2 based on a latent model structure, where the binary/

multinomial outcome results from discretizing a continuous latent variable that

is related to the predictors through a linear model. Their pseudo-R2 is defined

as the proportion of the variance of the latent variable that is explained by the

848 BO HU, JUN SHAO AND MARI PALTA

covariate. McFadden (1973) suggested an alternative, known as “likelihood-

ratio index”, comparing a model without any predictor to a model including all

predictors. It is defined as one minus the ratio of the log likelihood with inter-

cepts only, and the log likelihood with all predictors. If the slope parameters

are all 0, McFadden’s R2 is 0, but it is never 1. Maddala (1983) developed

another pseudo-R2 that can be applied to any model estimated by the maximum

likelihood method. This popular and widely used measure is expressed as

R2M = 1 − (

L(θ̃)

L(θ̂)

) 2

n

, (1)

where L(θ̃) is the maximized likelihood for the model without any predictor and

L(θ̂) is the maximized likelihood for the model with all predictors. In terms of

the likelihood ratio statistic λ = −2 log(L(θ̃)/L(θ̂)), R2M = 1 − e−λ/n. Maddala proved that R2M has an upper bound of 1 − (L(θ̃))2/n and, thus, suggested a normed measure based on a general principle of Cragg and Uhler (1970):

R2N = 1 −

(

L(θ̃)

L(θ̂)

) 2

n

1 − (L(θ̃)) 2

n

. (2)

While the statistics in (1) and (2) are widely used, their statistical properties

have not been fully investigated. Mittlböck and Schemper (1996) reviewed R2M and R2N along with other measures, but their results are mainly empirical and

numerical. The R2 for the linear model is interpreted as the proportion of the

variation in the response that can explained by the regressors. However, there is

no clear interpretation of the pseudo-R2s in terms of variance of the outcome in

logistic regression. Note that both R2M and R 2 N are statistics and thus random.

In linear regression, the standard R2 converges almost surely to the ratio of the

variability due to the covariates over the total variability as the sample size in-

creases to infinity. Once we know the limiting values of R2M and R 2 N , these limits

can be similarly used to understand how the pseudo-R2s capture the predictive

strength of the model. The pseudo-R2s for a given data set are point estimators

for the limiting values that are unknown. To account for the variability in esti-

mation, it is desirable to study the asymptotic sampling distributions of R2M and

R2N , which can be used to obtain asymptotic confidence intervals for the limiting

values of pseudo-R2s. Helland (1987) studied the sampling distributions of R2

statistics in linear regression.

In this article we study the behavior of R2M and R 2 N under the logistic re-

gression model. In Section 2, we derive the limits of R2M and R 2 N and provide

PSEUDO-R2 IN LOGISTIC REGRESSION MODEL 849

interpretations of them. We also present some graphs describing the behavior of

R2N across a range of practical situations. The asymptotic distributions of R 2 M

and R2N are derived in Section 3 and some simulation results are presented. An

example is given in Section 4.

2. What Does Pseudo-R2 Measure

In this section we explore the issue of what R2M in (1) and R 2 N in (2) measure

in the setting of binary or multinomial outcomes.

2.1. Limits of pseudo-R2s

Consider a study of n subjects whose outcomes fall in one of m categories.

Let Yi = (Yi1, . . . , Yim) ′ be the outcome vector associated with the ith subject,

where Yij = 1 if the outcome falls in the jth category, and Yij = 0 otherwise.

We assume that Y1, . . . , Yn are independent and that Yi is associated with a p-

dimensional vector Xi of predictors (covariates) through the multinomial logit

model

Pij = E(Yij|Xi) = exp(αj + X

′

iβj ) ∑m

k=1 exp(αk + X ′

iβk) , j = 1, . . . , m, (3)

where αm = βm = 0, α1, . . . , αm−1 are unknown scalar parameters, and β1, . . .,

βm−1 are unknown p-vectors of parameters. Let θ be the (p + 1)(m − 1) dimen- sional parameter (α1, β

′

1, . . . , αm−1, β ′

m−1). Then the likelihood function under

the multinomial logit model can be written as

L(θ) =

n ∏

i=1

P Yi1 i1 P

Yi2 i2 · · · P

Yim im . (4)

Procedures for obtaining the maximum likelihood estimator θ̂ of θ are available

in most statistical software packages. The following theorem provides the asymp-

totic limits of the pseudo-R2s defined in (1) and (2). Its proof is given in the

Appendix.

Theorem 1. Assume that covariates Xi, i = 1, . . . , n, are independent and

identically distributed random p-vectors with finite second moment. If

H1 = − m ∑

j=1

E(Pij ) log E(Pij ), (5)

H2 = − m ∑

j=1

E(Pij log Pij ), (6)

850 BO HU, JUN SHAO AND MARI PALTA

then, as n → ∞, R2M →p 1−e2(H2−H1) and R2N →p (1 − e2(H2−H1))/(1 − e−2H1 ), where →p denotes convergence in probability.

2.2. Interpretation of the limits of pseudo-R2s

It is useful to consider whether the limits of pseudo-R2 can be interpreted

much as R2 can be for linear regression analysis.

Theorem 1 reveals that both R2M and R 2 N converge to limits that can be

described in terms of entropy. If the covariates Xis are i.i.d., Yi = (Yi1, . . . , Yim) ′,

i = 1, . . . , n, are also i.i.d. multinomial distributed with probability vector (E(Pi1),

. . . , E(Pim)) where the expectation is taken over Xi. Then H1 given in (5) is

exactly the entropy measuring the marginal variation of Yi. Similarly, − ∑m

j=1 Pij log Pij corresponds to the conditional entropy measuring the variation of Yi given

Xi and H2 can be considered as the average conditional entropy. Therefore

H1 − H2 measures the difference in entropy explained by the covariate X, which is always greater than 0 by Jensen’s inequality, and is 0 if and only if the covariates

and outcomes are independent. For example, when (Xi, Yi) is bivariate normal,

H1 − H2 = log( √

1 − ρ2)−1 where ρ is the correlation coefficient, and the limit of R2M is ρ

2.

The limit of R2M is 1 − e−2(H1−H2) monotone in increasing H1 − H2. Then we can write the limit of R2N as the limit of R

2 M divided by its upper bound:

R2N →p 1 − e−2(H1−H2)

1 − e−2H1 =

e2H1 − e2H2 e2H1 − 1

.

When both H1 and H2 are small, 1 − e−2(H1−H2) ≈ 2(H1 − H2), 1 − e−2H1 ≈ 2H1 and the limit of R2N is approximately (H1 − H2)/H1, the entropy explained by the covariates relative to the marginal entropy H1.

2.3. Limits of R2 M

and R2 N

relative to model parameters

For illustration, we examine the magnitude of the limits of RM and RN under different parameter settings when the Xis are i.i.d. standard normal and

the outcome is binary. Figures 1 and 2 show the relationship between the limits

of R2M and R 2 N and the parameters α and β. In these figures, profile lines of the

limits are given for different levels of the response probability eα/(1 + eα) at the

mean of Xi and odds ratio e β per standard deviation of the covariate. The limits

tend to increase as the absolute value of β increases with other parameters fixed,

which is consistent with the behavior of the usual R2 in linear regression models.

However, we note that the limits tend to be low, even in models where the

parameters indicate a rather strong association with the outcome. For example,

PSEUDO-R2 IN LOGISTIC REGRESSION MODEL 851

a moderate size odds ratio of 2 per standard deviation of Xi is associated with

the limit of R2N at most 0.10. As the pseudo-R 2 measures do not correspond

in magnitude to what is familiar from R2 for ordinary regression, judgments

about the strength of the logistic model should refer to profiles such as those

provided in Figures 1 and 2. Knowing what odds ratio for a single predictor

model produces the same pseudo-R2 as a given multiple predictor model greatly

facilitates subject matter relevance assessment.

PSfrag replacements

0 2

4 6

0.0

0.2 0.4 0.6 0.8

e α

β

e α

1+eα

e β

Figure 1. Contour plot of limits of R2M against e α/(1 + eα) and odds ratio eβ .

852 BO HU, JUN SHAO AND MARI PALTA

PSfrag replacements

0 2

4 6

0.0

0.2 0.4 0.6 0.8

e

α

β

e α

1+eα

e β

Figure 2. Contour plot of limits of R2N against e α/(1 + eα) and odds ratio eβ.

It may be noted that neither R2N nor R 2 M can equal 1, except in degenerate

models. This property is a logical consequence of the nature of binary outcomes.

The denominator, 1 − (L(θ̃))2/n, equals the numerator when L(θ̂) equals 1, which occurs only for a degenerate outcome that is always 0 or 1. In fact, any perfectly

fitting model for binary data would predict probabilities that are only 0 or 1. This

constitutes a degenerate logistic model, which cannot be fit. In comparison to

PSEUDO-R2 IN LOGISTIC REGRESSION MODEL 853

the R2 for a linear model, R2 of 1 implies residual variance of 0. As the variance

and entropy of binomial and multinomial data depend on the mean, this again

can occur only when the predicted probabilities are 0 and 1. The mean-entropy

dependence influences the size of the pseudo-R2s and tends to keep them away

from 1 even when the mean probabilities are strongly dependent on the covariate.

For ease of model interpretation, investigators often categorize a continuous

variable, which leads to a loss of information. Consider a standard normally

distributed covariate. We calculate the limit of R2N when cutting the normal

covariate into two, three, five or six categories. The threshold points we choose

are 0 for two categories, ±1 for three categories, ±0.5, ±1 for five categories, and 0, ±0.5, ±1 for six categories. In Figures 3 and 4, we plot the corresponding limits of R2N against e

α/(1 + eα) by fixing β at 1, and against eβ by setting α = 1. The

fewer the number of categories we use for the covariate, the more information we

lose, i.e., the smaller the limit of R2N . In this example, we note that using five or

six categories retains most of the information provided by the original continuous

covariate.

PSfrag replacements

0 2

4

6

0.0

0.2 0.4

0.6

0.8

e α

β

e α

1+eα

eβ

0 .0

0 0 .0

5 0 .1

0 0 .1

5 0 .2

0 0 .2

5 0 .3

0

0.0 0.1

0.2

0.3

0.4

0.5

0.6 0.8

Continuous

Two

Three

Five

Six

Figure 3. Limit of R2N with covariate N (0, 1) dichotomized into K categories

against eα/(1 + eα), K = 2, 3, 5, 6

854 BO HU, JUN SHAO AND MARI PALTA

PSfrag replacements

0 2 4 6

0.0

0.2 0.4

0.6

0.8

e

α

β

e α

1+eα

eβ

0.00

0.05

0.10

0.15 0.20

0.25

0.30

0 .0

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6

0.8

Continuous

Two

Three

Five

Six

Figure 4. Limit of R2N with covariate N (0, 1) dichotomized into K categories

against odds ratio eβ , K = 2, 3, 5, 6

3. Sampling Distributions of Pseudo-R2s

The result in the previous section indicates that the limit of a pseudo-R2 is

a measure of the predictive strength of a model relating the logistic responses to

some predictors (covariates). The quantities R2M and R 2 N are statistics and are

random. They should be treated as estimators of their limiting values in assessing

the model strength. In this section, we derive the asymptotic distributions of R2M and R2N that are useful for deriving large sample confidence intervals.

3.1. Asymptotic distributions of pseudo-R2s

Theorem 2. Under the conditions of Theorem 1,

√ n [

R2M − (1 − e2(H2−H1)) ]

→d N (0, σ21 ) (7)

√ n

[

R2N − 1 − e2(H2−H1)

1 − e−2H1

]

→d N (0, σ22 ), (8)

PSEUDO-R2 IN LOGISTIC REGRESSION MODEL 855

where H1 and H2 are given by (5) and (6), σ 2 1 = g

′

1Σg1 and σ 2 2 = g

′

2Σg2 with

g1 = −2e2(H2−H1) (1 + log γ1, . . . , 1 + log γm, −1) , (9)

g2 = e−2H1 (1 − e2H2 )

(1 − e−2H1 )2 (

1 + log γ1, . . . , 1 + log γm, e 2H2

1 − e2H1 1 − e2H2

)

, (10)

Σ =

(

Cov(Yi) η

η′ �

)

. (11)

Here γj = Ex(Pij ), j = 1, . . . , m, is the expected probability that the outcome

falls in jth category, the jth element of η is ηj = Ex(Pij log Pij ) + γj H2, and

� = ∑m

j=1 Ex (

Pij (log Pij ) 2 )

− H22 . When all the slope parameters βj are 0 (i.e., Xi and Yi are uncorrelated),

both σ21 and σ 2 2 are zero. g1, g2 and Σ can be estimated by replacing the un-

known quantities, which are related to the covariate distribution, with consistent

estimators. For example, γ can be estimated by ( ∑

P̂i1/n, . . . , ∑

P̂im/n) ′.

Suppose gk, k = 1, 2, and Σ are estimated by ĝk and Σ̂, respectively. Theorem

2 leads to the following asymptotic 100(1 − α)% confidence interval for the limit of R2M :

(

R2M − Z α 2 ĝ′1Σ̂ĝ1, R

2 M + Z α

2 ĝ′1Σ̂ĝ1

)

, (12)

where Zα is the 1 − α quantile of the standard normal distribution. A confidence interval for the limit of R2N can be obtained by replacing R

2 M and ĝ1 in (12) with

R2N and ĝ2, respectively. If the resulting lower limit of the confidence interval is

below 0 or the upper limit is above 1, it is conventional to use the margin value

of 0 or 1.

3.2. Simulation results

In this section, we examine by simulation the finite sample performance of

the confidence intervals based on the asymptotic results derived in Section 3.1.

Our simulation experiments consider the logistic regression model with binary

outcome and a single normal covariate with mean 0 and standard deviation 1.

All the simulations were run with 3,000 replications of an artificially gener-

ated data set. In each replication, we simulated a sample of size 200 or 1,000

from the standard normal distribution as covariate vectors X , and simulated

200 or 1,000 binary outcomes according to success probability exp(α + βX)/(1+

exp(α + βX)). Tables 1 and 2 show the results for different values of α and β.

In all the simulations with sample size 1,000, the estimated confidence inter-

vals derived by Theorem 2 displayed coverage probability close to the expected

level of 0.95. Coverage probability is less satisfactory with sample size 200 when

the model is weak.

856 BO HU, JUN SHAO AND MARI PALTA

Table 1. Simulation average of pseudo-R2s and 95% confidence intervals in

the logit model with normal covariate (sample size=1,000).

α β R2M (limit) CI (coverage ∗) R2N (limit) CI (coverage

∗)

2 0.5 0.028 (0.027) (0.008,0.047) (0.930) 0.052 (0.050) (0.015,0.088)(0.930) 1 0.108 (0.103) (0.069,0.137) (0.937) 0.178 (0.178) (0.121,0.236) (0.940)

2 0.298 (0.298) (0.258,0.338) (0.929) 0.455 (0.454) (0.396,0.513) (0.927)

1 0.5 0.047 (0.046) (0.022,0.071) (0.937) 0.067 (0.066) (0.032,0.103) (0.940)

1 0.151 (0.150) (0.113,0.189) (0.943) 0.213 (0.213) (0.160,0.267) (0.945)

2 0.351 (0.350) (0.312,0.390) (0.922) 0.483 (0.483) (0.430,0.536) (0.929)

0.5 0.5 0.054 (0.053) (0.028,0.080) (0.940) 0.073 (0.072) (0.038,0.109) (0.941)

1 0.166 (0.166) (0.127,0.204) (0.948) 0.224 (0.226) (0.172,0.276) (0.948) 2 0.367 (0.365) (0.327,0.404) (0.931) 0.492 (0.491) (0.443,0.546) (0.933)

0 0.5 0.056 (0.055) (0.030,0.083) (0.938) 0.075 (0.074) (0.039,0.111) (0.938)

1 0.171 (0.171) (0.133,0.210) (0.931) 0.229 (0.228) (0.177,0.281) (0.933)

2 0.371 (0.370) (0.332,0.409) (0.925) 0.490 (0.494) (0.443,0.546) (0.929) ∗The relative frequency with which the intervals contain the true limit

Table 2. Simulation average of pseudo-R2s and 95% confidence intervals in

the logit model with normal covariate (sample size=200).

α β R2M (limit) CI (coverage) R 2 N (limit) CI (coverage)

2 0.5 0.031 (0.027) (0, 0.074) (0.912) 0.058 (0.050) (0, 0.138) (0.922)

1 0.107 (0.103) (0.0310, 0.182) (0.918) 0.185 (0.178) (0.0580, 0.312) (0.920)

2 0.299 (0.298) (0.2110, 0.387) (0.910) 0.458 (0.454) (0.3290, 0.588) (0.913)

1 0.5 0.050 (0.046) (0, 0.105) (0.915) 0.073 (0.066) (0, 0.151) (0.914)

1 0.152 (0.150) (0.0680, 0.235) (0.928) 0.215 (0.213) (0.0980, 0.332) (0.925)

2 0.351 (0.350) (0.2650, 0.437) (0.930) 0.484 (0.483) (0.3660, 0.601) (0.932)

0.5 0.5 0.058 (0.053) (0, 0.116) (0.919) 0.078 (0.072) (0, 0.157) (0.920) 1 0.168 (0.166) (0.0820, 0.253) (0.927) 0.227 (0.226) (0.1120, 0.343) (0.930)

2 0.367 (0.365) (0.2820, 0.452) (0.925) 0.494 (0.491) (0.3790, 0.608) (0.925)

0 0.5 0.059 (0.055) (0.0010, 0.118) (0.911) 0.079 (0.074) (0.0010, 0.158) (0.912)

1 0.174 (0.171) (0.0880, 0.260) (0.928) 0.233 (0.228) (0.1180, 0.347) (0.930)

2 0.372 (0.370) (0.2870, 0.457) (0.925) 0.496 (0.494) (0.3830, 0.610) (0.922)

4. Example

We now turn to an example of logistic regression from Fox’s (2001) text on

fitting generalized linear models. This example draws on data from the 1976 U.S.

Panel Study of Income Dynamics. There are 753 families in the data set with

8 variables. The variables are defined in Table 3. The logarithm of the wife’s

estimated wage rate is based on her actual earnings if she is in the labor force;

otherwise this variable is imputed from other predictors. The definition of other

variables is straightforward.

PSEUDO-R2 IN LOGISTIC REGRESSION MODEL 857

Table 3. Variables in the women labor force dataset.

Variable Description Remarks

lfp wife’s labor-force participation factor: no,yes k5 number of children ages 5 and younger 0-3, few 3’s

k618 number of children ages 6 to 18 0-8, few > 5

age wife’s age in years 30-60, single years

wc wife’s college attendance factor: no,yes hc husband’s college attendance factor: no,yes

lwg log of wife’s estimated wage rate see text

inc family income excluding wife’s income $1, 000s

We assume a binary logit model with no labor force participation as the

baseline category. Other variables are treated as predictors in the model. The

estimated model with all the predictors has the following form:

log P

1−P = 3.18−1.47k5−0.07k618−0.06age+0.81wc+0.11hc+0.61lwg−0.03inc,

where P is the probability that the wife in the family is in the labor force. The

variables k618 and hc are not statistically significant based on the likelihood-ratio

test. Table 4 shows the values of R2M and R 2 N , as well as 95% confidence intervals

of limits of R2M and R 2 N , for the model containg all the predictors, and models

excluding certain predictors.

Table 4. R2M and R 2 N with 95% confidence intervals of models for women

labor force data.

Model R2M (95% CI.) R 2 N (95% CI.)

Use all predictors 0.152 ( 0.109, 0.195) 0.205 ( 0.147, 0.262)

Exclude k5 0.074 ( 0.040, 0.108) 0.100 ( 0.054, 0.145)

Exclude age 0.123 ( 0.083, 0.164) 0.165 ( 0.111, 0.219)

Exclude wc 0.138 ( 0.096, 0.180) 0.185 ( 0.129, 0.241) Exclude lwg 0.133 ( 0.092, 0.174) 0.179 ( 0.123, 0.234)

Exclude inc 0.130 ( 0.087, 0.172) 0.175 ( 0.119, 0.230)

Exclude k618 0.151 ( 0.108, 0.194) 0.203 ( 0.145, 0.261)

Exclude hc 0.152 ( 0.109, 0.195) 0.204 ( 0.146, 0.262)

Use k618, hc only 0.003 (-0.005, 0.010) 0.004 (-0.006, 0.013)

For the model with all the covariates, R2M and R 2 N are around 0.15 and 0.20,

respectively. The results imply a moderately strong model when referencing the

odds ratio scale equivalents in Figure 1. Dropping a significant covariate results

in a notable decrease in the values of pseudo-R2s, while no significant change

occurs if we drop the insignificant covariates. R2M and R 2 N are near zero when we

858 BO HU, JUN SHAO AND MARI PALTA

exclude all the significant covariates. However, model selection procedures using

pseudo-R2 need further research.

Acknowledgements

The research work is supported by Grant CA-53786 from the National Cancer

Institute. The authors thank the referees and an editor for helpful comments.

Appendix

For the proof of results in Section 3, we begin with a lemma and then sketch

the main steps for Theorem 1 and 2.

Lemma 1. Assume that covariates Xi, i = 1, . . . , n, are i.i.d. random p-vectors

with finite second moment, then (log L(θ̂) − log L(θ))/ √

n →p 0, where θ̂ is the maximum likelihood estimator of θ.

Proof of Lemma 1. We first prove that ∂2 log L(θ)/∂θ∂θ ′

= Op(n). The score

function is

∂ log L(θ)

∂θ =

(

n ∑

i=1

(Yi1 − Pi1), n ∑

i=1

(Yi1 − Pi1)X′i, . . . , n ∑

i=1

(Yim − Pim)X′i

)

′

.

Let ηk = (αk, β ′

k) ′ ∈ Rp+1 for k = 1, . . . , m, and Ui = (1, X′i )′. Then

∂2 log L(θ)

∂ηk∂η ′

k

= − n ∑

i=1

Pik(1 − Pik)UiU ′i , k = 1, 2, . . . , m,

∂2 log L(θ)

∂ηk∂η ′

l

= − n ∑

i=1

PikPilUiU ′

i , k 6= l.

Since UiU ′

i =

(

1 X′i Xi XiX

′

i

)

, each element in the second derivative matrix

∂2 log L(θ)/∂θ∂θ ′

is Op(n) by assumption. For simplicity, we write this as

∂2 log L(θ)/∂θ∂θ ′

= Op(n). Let Sn(θ) = ∂ log L(θ)/∂θ, Jn(θ) =−∂2 log L(θ)/∂θ∂θ ′

and In(θ) = E(Jn(θ)), where the expectation is taken over covariates. It follows

that the cumulative information matrix In(θ) = Op(n). By a second-order Taylor

expansion,

log L(θ̂) − log L(θ)√ n

= Sn(θ̂)√

n

′

(θ̂ − θ) − 1

2 √

n (θ̂ − θ)′Jn(θ∗)(θ̂ − θ)

= (θ̂ − θ)′In(θ) 1

2 In(θ) −

1

2

√ n

Jn(θ ∗)

2n 3

2

√ nIn(θ)

− 1

2 In(θ) 1

2 (θ̂ − θ),

PSEUDO-R2 IN LOGISTIC REGRESSION MODEL 859

where θ∗ is a vector between θ and θ̂. The asymptotic normality results of the

MLE gives In(θ) 1/2(θ̂ −θ) → N (0, 1). The lemma then follows from the fact that

In(θ) −1/2

√ n = Op(1) and Jn(θ

∗)/n = Op(1).

Proof of Theorem 1. Let f (x) = log(1 − x)/2, then

f (R2) = 1

n log L(θ̃) − 1

n log L(θ̂)

= m ∑

j=1

nj n

log( nj n

) − 1 n

log L(θ) + 1

n

(

log L(θ) − log L(θ̂) )

.

The convergence of ∑m

j=1(nj /n) log(nj/n) and log L(θ)/n come from the Law

of Large Numbers. The results of the theorem follow from the lemma and the

Continuous Mapping Theorem.

Proof of Theorem 2. Let S2M = 1 − (L(θ̃)/L(θ))2/n and S2N = (1− (L(θ̃)/L(θ))2/n)/(1 − (L(θ̃))2/n). It follows from the lemma that S2M and S2N have the same asymptotic distribution as R2M and R

2 N , respectively, in the sense

that √

n(S2M − R2M ) →p 0 and √

n(S2N − R2N ) →p 0. Define Zi = (Yi1, . . . , Yim, Wi) where Wi =

∑m j=1 Yij log Pij . Then Zi’s form

a i.i.d. random sequence with µ = E(Zi) = (γ ′, ∑m

j=1 E(P1j log P1j )) = (γ ′, −E2),

Cov (Zi) = Σ, γ and Σ as defined in Section 3. By the Multidimensional Central

Limit Theorem,

√ n (

Z̄ − µ )

→ N (0, Σ). (13)

Let φ1(x1, . . . , xm) = 1 − e2( ∑

m j=1

xj log xj−xm+1) and φ2(x1, . . . , xm) = (1− e2( ∑

m j=1

xj log xj −xm+1))/(1 − e2 ∑

m j=1

xj log xj ). Applying the delta-method with φ1 and φ2 to (13), respectively, leads to the asymptotic normality results in

Theorem 2.

References

Agresti, A. (1986). Applying R2 type measures to ordered categorical data. Technometrics. 28,

133-138.

Ash, A. and Shwartz, M. (1999). R2: a useful measure of model performance when predicting

a dichotomous outcome. Statist. Medicine 18, 375-384.

Cox, D. R. and Wermuch, N. (1992). A comment on the coefficient of determination for binary

responses. Amer. Statist. 46, 1-4.

Cragg, J. G. and Uhler, R. S. (1970). The demand for automobiles. Canad. J. Economics 3,

386-406.

Draper, N. R. and Smith, H. (1998). Applied Rregression Analysis. 3rd edition. Wiley, New

York.

Fox, J. (2001). An R and S-Plus Companion to Applied Regression. Sage Publications.

860 BO HU, JUN SHAO AND MARI PALTA

Helland, I. S. (1987). On the interpretation and use of R2 in regression analysis, Biometrics 43,

61-69.

Laitila, T. (1993). A pseudo-R2 measure for limited and qualitative dependent variable models.

J. Econometrics 56, 341-356.

Long, J. S. (1997). Regression Models for Categorical and Limited Dependent Variables. Sage

Publications.

Maddala, G. S. (1983). Limited-Dependent and Qualitative Variables in Econometrics. Cam-

bridge University Press, Cambridge.

McFadden, D. (1973). Conditional logit analysis of qualitative choice behavior. In Frontiers in

Econometrics (Edited by P. Zarembka), 105-42. Academic Press, New York.

McKelvey, R. D. and Zavoina, W. (1975). A statistical model for the analysis of ordinal level

dependent variables. J. Math. Soc. 4, 103-120.

Mittlböck M. and Schemper, M. (1996). Explained variation for logistic regression. Statist.

Medicine 15, 1987-1997.

Nagelkerke, N. J. D. (1991). A note on a general definition of the coefficient of determination.

Biometrika 78, 691-693.

Zheng B. Y. and Agresti, A. (2000). Summarizing the predictive power of a generalized liner

model. Statist. Medicine 19, 1771-1781.

Department of Statistics, University of Wisconsin-Madison, Madison, WI, 53706, U.S.A.

E-mail: [email protected]

Department of Statistics, University of Wisconsin-Madison, Madison, WI, 53706, U.S.A.

E-mail: [email protected]

Department of Population Health Sciences, University of Wisconsin-Madison, Madison, WI,

53706, U.S.A.

E-mail: [email protected]

(Received August 2004; accepted July 2005)