Tag Archives: logistic regression

Veterinary Epidemiologic Research: GLM (part 4) – Exact and Conditional Logistic Regressions

Next topic on logistic regression: the exact and the conditional logistic regressions.

Exact logistic regression
When the dataset is very small or severely unbalanced, maximum likelihood estimates of coefficients may be biased. An alternative is to use exact logistic regression, available in R with the elrm package. Its syntax is based on an events/trials formulation. The dataset has to be collapsed into a data frame with unique combinations of predictors.
Another possibility is to use robust standard errors, and get comparable p-values to those obtained with exact logistic regression.

### exact logistic regression
x <- xtabs(~ casecont + interaction(dneo, dclox), data = nocardia)
x
        interaction(dneo, dclox)
casecont 0.0 1.0 0.1 1.1
       0  20  15   9  10
       1   2  44   3   5
> nocardia.coll <- data.frame(dneo = rep(1:0, 2), dclox = rep(1:0, each = 2),
+                    casecont = x[1, ], ntrials = colSums(x))
nocardia.coll
    dneo dclox casecont ntrials
0.0    1     1       20      22
1.0    0     1       15      59
0.1    1     0        9      12
1.1    0     0       10      15

library(elrm)
Le chargement a nécessité le package : coda
Le chargement a nécessité le package : lattice
mod5 <- elrm(formula = casecont/ntrials ~ dneo,
             interest = ~dneo,
             iter = 100000, dataset = nocardia.coll, burnIn = 2000)

### robust SE
library(robust)
Le chargement a nécessité le package : fit.models
Le chargement a nécessité le package : MASS
Le chargement a nécessité le package : robustbase
Le chargement a nécessité le package : rrcov
Le chargement a nécessité le package : pcaPP
Le chargement a nécessité le package : mvtnorm
Scalable Robust Estimators with High Breakdown Point (version 1.3-02)

mod6 <- glmrob(casecont ~ dcpct + dneo + dclox + dneo*dclox,
+                family = binomial, data = nocardia, method= "Mqle",
+                control= glmrobMqle.control(tcc=1.2))
> summary(mod6)

Call:  glmrob(formula = casecont ~ dcpct + dneo + dclox + dneo * dclox,      family = binomial, data = nocardia, method = "Mqle", control = glmrobMqle.control(tcc = 1.2)) 


Coefficients:
             Estimate Std. Error z-value Pr(>|z|)    
(Intercept) -4.440253   1.239138  -3.583 0.000339 ***
dcpct        0.025947   0.008504   3.051 0.002279 ** 
dneo         3.604941   1.034714   3.484 0.000494 ***
dclox        0.713411   1.193426   0.598 0.549984    
dneo:dclox  -2.935345   1.367212  -2.147 0.031797 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 
Robustness weights w.r * w.x: 
 89 weights are ~= 1. The remaining 19 ones are summarized as
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.1484  0.4979  0.6813  0.6558  0.8764  0.9525 

Number of observations: 108 
Fitted by method ‘Mqle’  (in 5 iterations)

(Dispersion parameter for binomial family taken to be 1)

No deviance values available 
Algorithmic parameters: 
   acc    tcc 
0.0001 1.2000 
maxit 
   50 
test.acc 
  "coef" 

Conditional logistic regression

Matched case-control studies analyzed with unconditional logistic regression model produce estimates of the odds ratios that are the square of their true value. But we can use conditional logistic regression to analyze matched case-control studies and get correct estimates. Instead of estimating a parameter for each matched set, a conditional model conditions the fixed effects out of the estimation. It can be run in R with clogit from the survival package:

temp <- tempfile()
> download.file(
+   "http://ic.upei.ca/ver/sites/ic.upei.ca.ver/files/ver2_data_R.zip", temp)
essai de l'URL 'http://ic.upei.ca/ver/sites/ic.upei.ca.ver/files/ver2_data_R.zip'
Content type 'application/zip' length 1107873 bytes (1.1 Mb)
URL ouverte
==================================================
downloaded 1.1 Mb

load(unz(temp, "ver2_data_R/sal_outbrk.rdata"))
unlink(temp)
### Salmonella outbreak dataset

library(survival)
mod7 <- clogit(casecontrol ~ slt_a + strata(match_grp), data = sal_outbrk)
summary(mod7)

Call:
coxph(formula = Surv(rep(1, 112L), casecontrol) ~ slt_a + strata(match_grp), 
    data = sal_outbrk, method = "exact")

  n= 112, number of events= 39 

        coef exp(coef) se(coef)     z Pr(>|z|)   
slt_a 1.4852    4.4159   0.5181 2.867  0.00415 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

      exp(coef) exp(-coef) lower .95 upper .95
slt_a     4.416     0.2265       1.6     12.19

Rsquare= 0.085   (max possible= 0.518 )
Likelihood ratio test= 10  on 1 df,   p=0.001568
Wald test            = 8.22  on 1 df,   p=0.004148
Score (logrank) test = 9.48  on 1 df,   p=0.002075
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Veterinary Epidemiologic Research: GLM – Evaluating Logistic Regression Models (part 3)

Third part on logistic regression (first here, second here).
Two steps in assessing the fit of the model: first is to determine if the model fits using summary measures of goodness of fit or by assessing the predictive ability of the model; second is to deterime if there’s any observations that do not fit the model or that have an influence on the model.

Covariate pattern
A covariate pattern is a unique combination of values of predictor variables.

mod3 <- glm(casecont ~ dcpct + dneo + dclox + dneo*dclox,
+             family = binomial("logit"), data = nocardia)
summary(mod3)

Call:
glm(formula = casecont ~ dcpct + dneo + dclox + dneo * dclox, 
    family = binomial("logit"), data = nocardia)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.9191  -0.7682   0.1874   0.5876   2.6755  

Coefficients:
                  Estimate Std. Error z value Pr(>|z|)    
(Intercept)      -3.776896   0.993251  -3.803 0.000143 ***
dcpct             0.022618   0.007723   2.928 0.003406 ** 
dneoYes           3.184002   0.837199   3.803 0.000143 ***
dcloxYes          0.445705   1.026026   0.434 0.663999    
dneoYes:dcloxYes -2.551997   1.205075  -2.118 0.034200 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 149.72  on 107  degrees of freedom
Residual deviance: 103.42  on 103  degrees of freedom
AIC: 113.42

Number of Fisher Scoring iterations: 5

library(epiR)
Package epiR 0.9-45 is loaded
Type help(epi.about) for summary information

mod3.mf <- model.frame(mod3)
(mod3.cp <- epi.cp(mod3.mf[-1]))
$cov.pattern
    id  n dcpct dneo dclox
1    1  7     0   No    No
2    2 38   100  Yes    No
3    3  1    25   No    No
4    4  1     1   No    No
5    5 11   100   No   Yes
8    6  1    25  Yes   Yes
10   7  1    14  Yes    No
12   8  4    75  Yes    No
13   9  1    90  Yes   Yes
14  10  1    30   No    No
15  11  3     5  Yes    No
17  12  9   100  Yes   Yes
22  13  2    20  Yes    No
23  14  8   100   No    No
25  15  2    50  Yes   Yes
26  16  1     7   No    No
27  17  4    50  Yes    No
28  18  1    50   No    No
31  19  1    30  Yes    No
34  20  1    99   No    No
35  21  1    99  Yes   Yes
40  22  1    80  Yes   Yes
48  23  1     3  Yes    No
59  24  1     1  Yes    No
77  25  1    10   No    No
84  26  1    83   No   Yes
85  27  1    95  Yes    No
88  28  1    99  Yes    No
89  29  1    25  Yes    No
105 30  1    40  Yes    No

$id
  [1]  1  2  3  4  5  1  1  6  5  7  5  8  9 10 11 11 12  1 12  1  5 13 14  2 15
 [26] 16 17 18  1  2 19  2 14 20 21 12 14  5  8 22 14  5  5  5  1 14 14 23  2 12
 [51] 14 12 11  5 15  2  8  2 24  2  2  8  2 17  2  2  2  2 12 12 12  2  2  2  5
 [76]  2 25  2 17  2  2  2  2 26 27 13 17 28 29 14  2 12  2  2  2  2  2  2  2  2
[101]  2  2  2  2 30  2  2  5

There are 30 covariate patterns in the dataset. The pattern dcpct=100, dneo=Yes, dclox=No appears 38 times.

Pearson and deviance residuals
Residuals represent the difference between the data and the model. The Pearson residuals are comparable to standardized residuals used for linear regression models. Deviance residuals represent the contribution of each observation to the overall deviance.

residuals(mod3) # deviance residuals
residuals(mod3, "pearson") # pearson residuals

Goodness-of-fit test
All goodness-of-fit tests are based on the premise that the data will be divided into subsets and within each subset the predicted number of outcomes will be computed and compared to the observed number of outcomes. The Pearson \chi^2 and the deviance \chi^2 are based on dividing the data up into the natural covariate patterns. The Hosmer-Lemeshow test is based on a more arbitrary division of the data.

The Pearson \chi^2 is similar to the residual sum of squares used in linear models. It will be close in size to the deviance, but the model is fit to minimize the deviance and not the Pearson \chi^2 . It is thus possible even if unlikely that the \chi^2 could increase as a predictor is added to the model.

sum(residuals(mod3, type = "pearson")^2)
[1] 123.9656
deviance(mod3)
[1] 103.4168
1 - pchisq(deviance(mod3), df.residual(mod3))
[1] 0.4699251

The p-value is large indicating no evidence of lack of fit. However, when using the deviance statistic to assess the goodness-of-fit for a nonsaturated logistic model, the \chi^2 approximation for the likelihood ratio test is questionable. When the covariate pattern is almost as large as N, the deviance cannot be assumed to have a \chi^2 distribution.
Now the Hosmer-Lemeshow test, usually dividing by 10 the data:

hosmerlem <- function (y, yhat, g = 10) {
+   cutyhat <- cut(yhat, breaks = quantile(yhat, probs = seq(0, 1, 1/g)),
+                  include.lowest = TRUE)
+   obs <- xtabs(cbind(1 - y, y) ~ cutyhat)
+   expect <- xtabs(cbind(1 - yhat, yhat) ~ cutyhat)
+   chisq <- sum((obs - expect)^2 / expect)
+   P <- 1 - pchisq(chisq, g - 2)
+   c("X^2" = chisq, Df = g - 2, "P(>Chi)" = P)
+ }
hosmerlem(y = nocardia$casecont, yhat = fitted(mod3))
Erreur dans cut.default(yhat, breaks = quantile(yhat, probs = seq(0, 1, 1/g)),  (depuis #2) :
  'breaks' are not unique

The model used has many ties in its predicted probabilities (too few covariate values?) resulting in an error when running the Hosmer-Lemeshow test. Using fewer cut-points (g = 5 or 7) does not solve the problem. This is a typical example when not to use this test. A better goodness-of-fit test than Hosmer-Lemeshow and Pearson / deviance \chi^2 tests is the le Cessie – van Houwelingen – Copas – Hosmer unweighted sum of squares test for global goodness of fit (also here) implemented in the rms package (but you have to implement your model with the lrm function of this package):

mod3b <- lrm(casecont ~ dcpct + dneo + dclox + dneo*dclox, nocardia,
+              method = "lrm.fit", model = TRUE, x = TRUE, y = TRUE,
+              linear.predictors = TRUE, se.fit = FALSE)
residuals(mod3b, type = "gof")
Sum of squared errors     Expected value|H0                    SD 
           16.4288056            16.8235055             0.2775694 
                    Z                     P 
           -1.4219860             0.1550303 

The p-value is 0.16 so there’s no evidence the model is incorrect. Even better than these tests would be to check for linearity of the predictors.

Overdispersion
Sometimes we can get a deviance that is much larger than expected if the model was correct. It can be due to the presence of outliers, sparse data or clustering of data. The approach to deal with overdispersion is to add a dispersion parameter \sigma^2 . It can be estimated with: \hat{\sigma}^2 = \frac{\chi^2}{n - p} (p = probability of success). A half-normal plot of the residuals can help checking for outliers:

library(faraway)
halfnorm(residuals(mod1))
Half-normal plot of the residuals
Half-normal plot of the residuals

The dispesion parameter of model 1 can be found as:

(sigma2 <- sum(residuals(mod1, type = "pearson")^2) / 104)
[1] 1.128778
drop1(mod1, scale = sigma2, test = "F")
Single term deletions

Model:
casecont ~ dcpct + dneo + dclox

scale:  1.128778 

       Df Deviance    AIC F value    Pr(>F)    
<none>      107.99 115.99                      
dcpct   1   119.34 124.05 10.9350  0.001296 ** 
dneo    1   125.86 129.82 17.2166 6.834e-05 ***
dclox   1   114.73 119.96  6.4931  0.012291 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 
Message d'avis :
In drop1.glm(mod1, scale = sigma2, test = "F") :
  le test F implique une famille 'quasibinomial'

The dispersion parameter is not very different than one (no dispersion). If dispersion was present, you could use it in the F-tests for the predictors, adding scale to drop1.

Predictive ability of the model
A ROC curve can be drawn:

predicted <- predict(mod3)
library(ROCR)
prob <- prediction(predicted, nocardia$casecont, 
+                    label.ordering = c('Control', 'Case'))
tprfpr <- performance(prob, "tpr", "fpr")
tpr <- unlist(slot(tprfpr, "y.values"))
fpr <- unlist(slot(tprfpr, "x.values"))
roc <- data.frame(tpr, fpr)
ggplot(roc) + geom_line(aes(x = fpr, y = tpr)) + 
+   geom_abline(intercept = 0, slope = 1, colour = "gray") + 
+   ylab("Sensitivity") + 
+   xlab("1 - Specificity")
ROC curve
ROC curve

Identifying important observations
Like for linear regression, large positive or negative standardized residuals allow to identify points which are not well fit by the model. A plot of Pearson residuals as a function of the logit for model 1 is drawn here, with bubbles relative to size of the covariate pattern. The plot should be an horizontal band with observations between -3 and +3. Covariate patterns 25 and 26 are problematic.

nocardia$casecont.num <- as.numeric(nocardia$casecont) - 1
mod1 <- glm(casecont.num ~ dcpct + dneo + dclox, family = binomial("logit"),
+             data = nocardia) # "logit" can be omitted as it is the default
mod1.mf <- model.frame(mod1)
mod1.cp <- epi.cp(mod1.mf[-1])
nocardia.cp <- as.data.frame(cbind(cpid = mod1.cp$id,
+                                    nocardia[ , c(1, 9:11, 13)],
+                                    fit = fitted(mod1)))
### Residuals and delta betas based on covariate pattern:
mod1.obs <- as.vector(by(as.numeric(nocardia.cp$casecont.num),
+                          as.factor(nocardia.cp$cpid), FUN = sum))
mod1.fit <- as.vector(by(nocardia.cp$fit, as.factor(nocardia.cp$cpid),
+                          FUN = min))
mod1.res <- epi.cpresids(obs = mod1.obs, fit = mod1.fit,
+                          covpattern = mod1.cp)

mod1.lodds <- as.vector(by(predict(mod1), as.factor(nocardia.cp$cpid),
+                            FUN = min))

plot(mod1.lodds, mod1.res$spearson,
+      type = "n", ylab = "Pearson Residuals", xlab = "Logit")
text(mod1.lodds, mod1.res$spearson, labels = mod1.res$cpid, cex = 0.8)
symbols(mod1.lodds, mod1.res$pearson, circles = mod1.res$n, add = TRUE)
Bubble plot of standardized residuals
Bubble plot of standardized residuals

The hat matrix is used to calculate leverage values and other diagnostic parameters. Leverage measures the potential impact of an observation. Points with high leverage have a potential impact. Covariate patterns 2, 14, 12 and 5 have the largest leverage values.

mod1.res[sort.list(mod1.res$leverage, decreasing = TRUE), ]
cpid  leverage
2   0.74708052
14  0.54693851
12  0.54017700
5   0.42682684
11  0.21749664
1   0.19129427
...

Delta-betas provides an overall estimate of the effect of the j^{th} covariate pattern on the regression coefficients. It is analogous to Cook’s distance in linear regression. Covariate pattern 2 has the largest delta-beta (and represents 38 observations).

mod1.res[sort.list(mod1.res$sdeltabeta, decreasing = TRUE), ]
cpid sdeltabeta
2    7.890878470
14   3.983840529
...

Veterinary Epidemiologic Research: GLM – Logistic Regression (part 2)

Second part on logistic regression (first one here).
We used in the previous post a likelihood ratio test to compare a full and null model. The same can be done to compare a full and nested model to test the contribution of any subset of parameters:

mod1.nest <- glm(casecont ~ dcpct, family = binomial("logit"),
+             data = nocardia)
lr.mod1.nest <- -(deviance(mod1.nest) / 2)
(lr <- 2 * (lr.mod1 - lr.mod1.nest))
[1] 30.16203
1 - pchisq(lr, 2)
[1] 2.820974e-07
### or, more straightforward, using anova to compare the 2 models:
anova(mod1, mod1.nest, test = "Chisq")
Analysis of Deviance Table

Model 1: casecont ~ dcpct + dneo + dclox
Model 2: casecont ~ dcpct
  Resid. Df Resid. Dev Df Deviance  Pr(>Chi)    
1       104     107.99                          
2       106     138.15 -2  -30.162 2.821e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Interpretation of coefficients

### example 16.2
nocardia$dbarn <- as.factor(nocardia$dbarn)
mod2 <- glm(casecont ~ dcpct + dneo + dclox + dbarn,
+               family = binomial("logit"), data = nocardia)
(mod2.sum <- summary(mod2))

Call:
glm(formula = casecont ~ dcpct + dneo + dclox + dbarn, family = binomial("logit"), 
    data = nocardia)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.6949  -0.7853   0.1021   0.7692   2.6801  

Coefficients:
               Estimate Std. Error z value Pr(>|z|)    
(Intercept)   -2.445696   0.854328  -2.863  0.00420 ** 
dcpct          0.021604   0.007657   2.821  0.00478 ** 
dneoYes        2.685280   0.677273   3.965 7.34e-05 ***
dcloxYes      -1.235266   0.580976  -2.126  0.03349 *  
dbarntiestall -1.333732   0.631925  -2.111  0.03481 *  
dbarnother    -0.218350   1.154293  -0.189  0.84996    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 149.72  on 107  degrees of freedom
Residual deviance: 102.32  on 102  degrees of freedom
AIC: 114.32

Number of Fisher Scoring iterations: 5

cbind(exp(coef(mod2)), exp(confint(mod2)))
Waiting for profiling to be done...
                               2.5 %     97.5 %
(Intercept)    0.08666577 0.01410982  0.4105607
dcpct          1.02183934 1.00731552  1.0383941
dneoYes       14.66230075 4.33039752 64.5869271
dcloxYes       0.29075729 0.08934565  0.8889877
dbarntiestall  0.26349219 0.06729031  0.8468235
dbarnother     0.80384385 0.08168466  8.2851189

Note: Dohoo do not report the profile likelihood-based confidence interval on page 404. As said in the previous post, the profile likelihood-based CI is preferable due to the Hauck-Donner effect (overestimation of the SE).

Using neomycin in the herd increases the log odds of Nocardia mastitis by 2.7 units (or using neomycin increases the odds 14.7 times). Since Nocardia mastitis is a rare condition, we can interpret the odds ratio as a risk ratio (RR) and say neomycin increases the risk of Nocardia mastitis by 15 times. Changing the percentage of dry cows treated from 50 to 75% increases the log odds of disease by (75 - 50) \times 0.022 = 0.55 units (or 1.022^{(75-50)} = 1.72 ). An increase of 25% in the percentage of cows dry-treated increases the risk of disease by about 72% (i.e. 1.72 times). Tiestall barns and other barn types both have lower risks of Nocardia mastitis than freestall barns.

The significance of the main effects can be tested with:

drop1(mod2, test = "Chisq")
Single term deletions

Model:
casecont ~ dcpct + dneo + dclox + as.factor(dbarn)
                 Df Deviance    AIC     LRT  Pr(>Chi)    
<none>                102.32 114.32                      
dcpct             1   111.36 121.36  9.0388  0.002643 ** 
dneo              1   123.91 133.91 21.5939 3.369e-06 ***
dclox             1   107.02 117.02  4.7021  0.030126 *  
as.factor(dbarn)  2   107.99 115.99  5.6707  0.058697 .  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

The drop1 function tests each predictor relative to the full model.

Presenting effects of factors on the probability scale
Usually we think about the probability of disease rather than the odds, and the probability of disease is not linearly related to the factor of interest. A unit increase in the factor does not increase the probability of disease by a fixed amount. It depends on the level of the factor and the levels of other factors in the model.

mod1 <- glm(casecont ~ dcpct + dneo + dclox, family = binomial("logit"),
+             data = nocardia)
nocardia$neo.pred <- predict(mod1, type = "response", se.fit = FALSE)
library(ggplot2)
ggplot(nocardia, aes(x = dcpct, y = neo.pred, group = dneo,
+                      colour = factor(dneo))) + 
+   stat_smooth(method = "glm", family = "binomial", se = FALSE) +
+   labs(colour = "Neomycin", x = "Percent of cows dry treated",
+        y = "Probability of Nocardia")
Effect of dry-cow treatment
Effect of dry-cow treatment

Next: evaluating logistic regression models.