This page shows an example of logistic regression with footnotes explaining the output. The data were collected on 200 high school students, with measurements on various tests, including science, math, reading and social studies. The response variable is high writing test score (honcomp), where a writing score greater than or equal to 60 is considered high, and less than 60 considered low; from which we explore its relationship with gender (female), reading test score (read), and science test score (science). The dataset used in this page can be downloaded from http://www.ats.ucla.edu/stat/sas/webbooks/reg/default.htm.
data logit; set "c:\temp\hsb2"; honcomp = (write >= 60); run; proc logistic data= logit descending; model honcomp = female read science; run; The LOGISTIC Procedure Model Information Data Set WORK.LOGIT Response Variable honcomp Number of Response Levels 2 Model binary logit Optimization Technique Fisher's scoring Number of Observations Read 200 Number of Observations Used 200 Response Profile Ordered Total Value honcomp Frequency 1 1 53 2 0 147 Probability modeled is honcomp=1. Model Convergence Status Convergence criterion (GCONV=1E-8) satisfied. Model Fit Statistics Intercept Intercept and Criterion Only Covariates AIC 233.289 168.236 SC 236.587 181.430 -2 Log L 231.289 160.236 Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 71.0525 3 <.0001 Score 58.6092 3 <.0001 Wald 39.8751 3 <.0001 Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -12.7772 1.9759 41.8176 <.0001 female 1 1.4825 0.4474 10.9799 0.0009 read 1 0.1035 0.0258 16.1467 <.0001 science 1 0.0948 0.0305 9.6883 0.0019 Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits female 4.404 1.832 10.584 read 1.109 1.054 1.167 science 1.099 1.036 1.167 Association of Predicted Probabilities and Observed Responses Percent Concordant 85.6 Somers' D 0.714 Percent Discordant 14.2 Gamma 0.715 Percent Tied 0.2 Tau-a 0.279 Pairs 7791 c 0.857
Model Information Data Seta WORK.LOGIT Response Variableb honcomp Number of Response Levelsc 2 Modeld binary logit Optimization Techniquee Fisher's scoring Number of Observations Readf 200 Number of Observations Usedf 200 Response Profile Ordered Total Valueg honcompg Frequencyh 1 1 53 2 0 147 Probability modeled is honcomp=1.i
a. Data Set - This the data set used in this procedure.
b. Response Variable - This is the response variable in the logistic regression.
c. Number of Response Levels - This is the number of levels our response variable has.
d. Model - This is the type of regression model that was fit to our data. The term logit and logistic are exchangeable.
e. Optimization Technique - This refers to the iterative method of estimating the regression parameters. In SAS, the default is method is Fisher's scoring method, whereas in Stata, it is the Newton-Raphson algorithm. Both techniques yield the same estimate for the regression coefficient; however, the standard errors differ between the two methods. For further discussion, see Regression Models for Categorical and Limited Dependent Variables by J. Scott Long (page 56).
f. Number of Observations Read and Number of Observations Used - This is the number of observations read and the number of observation used in the analysis. The Number of Observations Used may be less than the Number of Observations Read if there are missing values for any variables in the equation. By default, SAS does a listwise deletion of incomplete cases.
g. Ordered Value and honcomp - Ordered value refers to how SAS orders/models the levels of the dependent variable. When we specified the descending option in the procedure statement, SAS treats the levels of honcomp in a descending order (high to low), such that when the logit regression coefficients are estimated, a positive coefficient corresponds to a positive relationship for high write status, and a negative coefficient has a negative relationship with high write status. Special attention needs to be placed on the ordered value since it can lead to erroneous interpretation. By default SAS models the 0's, whereas most other statistics packages model the 1's. The descending option is necessary so that SAS models the 1's.
h. Total Frequency - This is the frequency distribution of the response variable. Our response variable has 53 observations with a high write score and 147 with a low write score.
i. Probability modeled is honcomp=1 - This is a note informing which level of the response variable we are modeling. See superscript g for further discussion of the descending option and its influence on which level of the response variable is being modeled.
Model Convergence Statusj Convergence criterion (GCONV=1E-8) satisfied. Model Fit Statistics Intercept Intercept and Criterionk Onlyl Covariatesm AIC 233.289 168.236 SC 236.587 181.430 -2 Log L 231.289 160.236 Testing Global Null Hypothesis: BETA=0 Testn Chi-Squareo DFo Pr > ChiSqo Likelihood Ratio 71.0525 3 <.0001 Score 58.6092 3 <.0001 Wald 39.8751 3 <.0001
j. Model Convergence Status - This describes whether the maximum-likehood algorithm has converged or not, and what kind of convergence criterion is used to asses convergence. The default criterion is the relative gradient convergence criterion (GCONV), and the default precision is 10-8.
k. Criterion - Underneath are various measurements used to assess the model fit. The first two, Akaike Information Criterion (AIC) and Schwarz Criterion (SC) are deviants of negative two times the Log-Likelihood (-2 Log L). AIC and SC penalize the log-likelihood by the number of predictors in the model.
AIC - This is the Akaike Information Criterion. It is calculated as AIC = -2 Log L + 2((k-1) + s), where k is the number of levels of the dependent variable and s is the number of predictors in the model. AIC is used for the comparison of nonnested models on the same sample. Ultimately, the model with the smallest AIC is considered the best, although the AIC value itself is not meaningful.
SC - This is the Schwarz Criterion. It is defined as - 2 Log L + ((k-1) + s)*log(Σ fi), where fi's are the frequency values of the ith observation, and k and s were defined previously. Like AIC, SC penalizes for the number of predictors in the model and the smallest SC is most desirable and the value itself is not meaningful..
-2 Log L - This is negative two times the log-likelihood. The -2 Log L is used in hypothesis tests for nested models and the value in itself is not meaningful.
l. Intercept Only - This column refers to the respective criterion statistics with no predictors in the model, i.e., just the response variable.
m. Intercept and Covariates - This column corresponds to the respective criterion statistics for the fitted model. A fitted model includes all independent variables and the intercept. We can compare the values in this column with the criteria corresponding Intercept Only value to assess model fit/significance.
n. Test - These are three asymptotically equivalent Chi-Square tests. They test against the null hypothesis that at least one of the predictors' regression coefficient is not equal to zero in the model. The difference between them are where on the log-likelihood function they are evaluated. For further discussion, see Categorical Data Analysis, Second Edition, by Alan Agresti (pages 11-13).
Likelihood Ratio - This is the Likelihood Ratio (LR) Chi-Square test that at least one of the predictors' regression coefficient is not equal to zero in the model. The LR Chi-Square statistic can be calculated by -2 Log L(null model) - 2 Log L(fitted model) = 231.289-160.236 = 71.05, where L(null model) refers to the Intercept Only model and L(fitted model) refers to the Intercept and Covariates model.
Score - This is the Score Chi-Square Test that at least one of the predictors' regression coefficient is not equal to zero in the model.
Wald - This is the Wald Chi-Square Test that at least one of the predictors' regression coefficient is not equal to zero in the model.
o. Chi-Square, DF and Pr > ChiSq - These are the Chi-Square test statistic, Degrees of Freedom (DF) and associated p-value (PR>ChiSq) corresponding to the specific test that all of the predictors are simultaneously equal to zero. We are testing the probability (PR>ChiSq) of observing a Chi-Square statistic as extreme as, or more so, than the observed one under the null hypothesis; the null hypothesis is that all of the regression coefficients in the model are equal to zero. The DF defines the distribution of the Chi-Square test statistics and is defined by the number of predictors in the model. Typically, PR>ChiSq is compared to a specified alpha level, our willingness to accept a type I error, which is often set at 0.05 or 0.01. The small p-value from the all three tests would lead us to conclude that at least one of the regression coefficients in the model is not equal to zero.
Analysis of Maximum Likelihood Estimates Standard Wald Parameterp DFq Estimater Errors Chi-Squaret Pr > ChiSqt Intercept 1 -12.7772 1.9759 41.8176 <.0001 female 1 1.4825 0.4474 10.9799 0.0009 read 1 0.1035 0.0258 16.1467 <.0001 science 1 0.0948 0.0305 9.6883 0.0019 Odds Ratio Estimates Point 95% Wald Effectu Estimatev Confidence Limitsw female 4.404 1.832 10.584 read 1.109 1.054 1.167 science 1.099 1.036 1.167
p. Parameter - Underneath are the predictor variables in the model and the intercept.
q. DF - This column gives the degrees of freedom corresponding to the Parameter. Each Parameter estimated in the model requires one DF and defines the Chi-Square distribution to test whether the individual regression coefficient is zero, given the other variables are in the model.
r. Estimate - These are the binary logit regression estimates for the
Parameters in the model. The logistic regression model models the log
odds of a positive response (probability modeled is honcomp=1) as a linear
combination the predictor variables. This is written as
log[ p / (1-p) ] = b0 + b1*female + b2*read + b3 *science,
where p is the probability that honcomp is 1. For our model, we have,
log[ p / (1-p) ] = -12.78 + 1.48*female + 0.10*read + 0.09*science.
We can interpret the parameter estimates as follows: for a one unit change in the predictor variable, the difference in log-odds for a positive outcome is expected to change by the respective coefficient, given the other variables in the model are held constant.
Intercept - This is the logistic regression
estimate when all variables in the model are evaluated at zero. For males (the
variable female evaluated at zero) with zero read and science
test scores, the log-odds for high write score is -12.777. Note that evaluating
read and science at zero is out of the range of plausible test
scores. If the test scores were mean-centered, the intercept would have a
natural interpretation: the expected log-odds for high write score for males
with an average read and science test score.
female - This is the estimated logistic regression coefficient comparing females to males, given the other variables are held constant in the model. The difference in log-odds is expected to be 1.4825 units higher for females compared to males, while holding the other variables constant in the model.
read - This is the estimate logistic regression coefficient for a one unit change in read score, given the other variables in the model are held constant. If a student were to increase her read score by one point, her difference in log-odds for high write score is expected to increase by 0.10 unit, given the other variables in the model are held constant.
science - This is the estimate logistic regression coefficient for a one unit change in science score, given the other variables in the model are held constant. If a student were to increase her science score by one point, the difference in log-odds for high write score is expected to increase by 0.095 unit, given the other variables in the model are held constant.
s. Standard Error - These are the standard errors of the individual regression coefficients. They are used in both the 95% Wald Confidence Limits, superscript w, and the Chi-Square test statistic, superscript t.
t. Chi-Square and Pr > ChiSq - These are the test statistics and p-values, respectively, testing the null hypothesis that an individual predictor's regression coefficient is zero, given the other predictor variables are in the model. The Chi-Square test statistic is the squared ratio of the Estimate to the Standard Error of the respective predictor. The Chi-Square value follows a central Chi-Square distribution with degrees of freedom given by DF, which is used to test against the alternative hypothesis that the Estimate is not equal to zero. The probability that a particular Chi-Square test statistic is as extreme as, or more so, than what has been observed under the null hypothesis is defined by Pr>ChiSq.
u. Effect - Underneath are the predictor variables that are interpreted in terms of odds ratios.
v. Point Estimate - Underneath are the odds ratio corresponding to Effect. The odds ratio is obtained by exponentiating the Estimate, exp[Estimate]. The difference in the log of two odds is equal to the log of the ratio of these two odds. The log of the ratio of two odds is the log odds ratio. Hence, the interpretation of Estimate--the coefficient was interpreted as the difference in log-odds--could also be done in terms of log-odds ratio. When the Estimate is exponentiated, the log-odds ratio becomes the odds ratio. We can interpret the odds ratio as follows: for a one unit change in the predictor variable, the odds ratio for a positive outcome is expected to change by the respective coefficient, given the other variables in the model are held constant.
w. 95% Wald Confidence Limits - This is the Wald Confidence Interval (CI) of an individual odds ratio, given the other predictors are in the model. For a given predictor variable with a level of 95% confidence, we'd say that we are 95% confident that upon repeated trials, 95% of the CI's would include the "true" population odds ratio. The CI is equivalent to the Chi-Square test statistic: if the CI includes one, we'd fail to reject the null hypothesis that a particular regression coefficient equals zero and the odds ratio equals one, given the other predictors are in the model. An advantage of a CI is that it is illustrative; it provides information on where the "true" parameter may lie and the precision of the point estimate for the odds ratio.
Association of Predicted Probabilities and Observed Responses Percent Concordantx 85.6 Somers' Dbb 0.714 Percent Discordanty 14.2 Gammacc 0.715 Percent Tiedz 0.2 Tau-add 0.279 Pairsaa 7791 cee 0.857
x. Percent Concordant - A pair of observations with different observed responses is said to be concordant if the observation with the lower ordered response value (honcomp = 0) has a lower predicted mean score than the observation with the higher ordered response value (honcomp = 1). See Pairs, superscript aa, for what defines a pair.
y. Percent Discordant - If the observation with the lower ordered response value has a higher predicted mean score than the observation with the higher ordered response value, then the pair is discordant.
z. Percent Tied - If a pair of observations with different responses is neither concordant nor discordant, it is a tie.
aa. Pairs - This is the total number of distinct pairs in which one case has an observed outcome different from the other member of the pair. In the Response Profile table in the Model Information section above, we see that there are 53 observations with honcomp=1 and 147 observations with honcomp=0. Thus the total number of pairs with different outcomes is 53*147=7791.
bb. Somers' D - Somer's D is used to determine the strength and direction of relation between pairs of variables. Its values range from -1.0 (all pairs disagree) to 1.0 (all pairs agree). It is defined as (nc-nd)/t where nc is the number of pairs that are concordant, nd the number of pairs that are discordant, and t is the number of total number of pairs with different responses. In our example, it equals the difference between the percent concordant and the percent discordant divided by 100: (85.6-14.2)/100 = 0.714.
cc. Gamma - The Goodman-Kruskal Gamma method does not penalize for ties on either variable. Its values range from -1.0 (no association) to 1.0 (perfect association). Because it does not penalize for ties, its value will generally be greater than the values for Somer's D.
dd. Tau-a - Kendall's Tau-a is a modification of Somer's D that takes into the account the difference between the number of possible paired observations and the number of paired observations with a different response. It is defined to be the ratio of the difference between the number of concordant pairs and the number of discordant pairs to the number of possible pairs (2(nc-nd)/(N(N-1)). Usually Tau-a is much smaller than Somer's D since there would be many paired observations with the same response.
ee. c - c is equivalent to the well known measure ROC. c ranges from 0.5 to 1, where 0.5 corresponds to the model randomly predicting the response, and a 1 corresponds to the model perfectly discriminating the response.
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