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Plan of the day

In today’s lecture we continue our discussion of the multiple logistic regression model

Topics covered

- Models and submodels
- Residuals for Multiple Logistic Regression
- Diagnostics in Multiple Logistic Regression
- No analogue of R2

Reference: Coursebook, sections 5.2.3, 5.2.3

Comparison of models

- Suppose model 1 and model 2 are two models, with model 2 a submodel of model1
- If Model 2 is in fact correct, then the difference in the deviances will have approximately a chi-squared distribution
- df equals the difference in df of the separate models
- Approximation OK for grouped and ungrouped data

Example: kyphosis data

- Is age alone an adequate model?

> age.glm<-glm(Kyphosis~Age+I(Age^2),family=binomial, data=kyphosis.df)

Null deviance: 83.234 on 80 degrees of freedom

Residual deviance: 72.739 on 78 degrees of freedom

AIC: 78.739

Full model has deviance 54.428 on 76 df

Chisq is 72.739 - 54.428 = 18.311 on 78-76=2 df

> 1-pchisq(18.311,2)

[1] 0.0001056372

Highly significant: need at least one of start and number

Anova in R

Two-model form: comparing

> anova(age.glm,kyphosis.glm, test=“Chi”)

Analysis of Deviance Table

Model 1: Kyphosis ~ Age + I(Age^2)

Model 2: Kyphosis ~ Age + I(Age^2) + Start + Number

Resid. Df Resid. Dev Df Deviance P(>|Chi|)

1 78 72.739

2 76 54.428 2 18.311 0.0001056 ***

Residuals

- Two kinds of residuals
- Pearson residuals
- useful for grouped data only
- similar to residuals in linear regression, actual minus fitted value
- Deviance residuals
- useful for grouped and ungrouped data
- Measure contribution of each covariate pattern to the deviance

Pearson residuals

Pearson residual for pattern i is

Probability predicted by model

Standardized to have approximately unit variance, so big if more than 2 in absolute value

Deviance residuals (i)

- For grouped data, the deviance is

Deviance residuals (i)

- Thus, the deviance can be written as the sum of squares of M quantities d1, …, dM ,one for each covariate pattern
- Each di is the contribution to the deviance from the ith covariate pattern
- If deviance residual is big (more than about 2 in magnitude), then the covariate pattern has a big influence on the likelihood, and hence the estimates

Calculating residuals

> pearson.residuals<-residuals(budworm.glm,

type="pearson")

- > deviance.residuals<-residuals(budworm.glm, type="deviance")
- > par(mfrow=c(1,2))

> plot(pearson.residuals, ylab="residuals",

main="Pearson")

- > abline(h=0,lty=2)

> plot(deviance.residuals, ylab="residuals", main="Deviance")

- > abline(h=0,lty=2)

Diagnostics: outlier detection

- Large residuals indicate covariate patterns poorly fitted by the model
- Large Pearson residuals indicate a poor match between the “maximum model probabilities” and the logistic model probabilities, for grouped data
- Large deviance residuals indicate influential points
- Example: budworm data

Diagnostics: detecting non-linear regression functions

- For a single x, plot the logits of the maximal model probabilities against x
- For multiple x’s, plot Pearson residuals against fitted probabilities, against individual x’s
- If the data has most ni’s equal to 1, so can’t be grouped, try gam (cf kyphosis data)

Example: budworms

- Plot Pearson residuals versus dose, plot shows a curve

Diagnostics: influential points

Will look at 3 diagnostics

- Hat matrix diagonals
- Cook’s distance
- Leave-one-out Deviance Change

Example: vaso-constriction data

Data from study of reflex vaso-constriction (narrowing of the blood vessels) of the skin of the fingers

- Can be caused caused by sharp intake of breath

Example: vaso-constriction data

Variables measured:

Response = 0/1

1=vaso-constriction occurs, 0 = doesn’t occur

Volume: volume of air breathed in

Rate: rate of intake of breath

Data

- Volume Rate Response
- 1 3.70 0.825 1
- 2 3.50 1.090 1
- 3 1.25 2.500 1
- 4 0.75 1.500 1
- 5 0.80 3.200 1
- 6 0.70 3.500 1
- 7 0.60 0.750 0
- 8 1.10 1.700 0
- 9 0.90 0.750 0
- 10 0.90 0.450 0
- 11 0.80 0.570 0
- 12 0.55 2.750 0
- 0.60 3.000 0
- . . . 39 obs in all

Plot of data

> plot(Rate,Volume,type="n", cex=1.2)

> text(Rate,Volume,1:39,

col=ifelse(Response==1, “red",“blue"),

cex=1.2)

> text(2.3,3.5,“blue: no VS", col=“blue",

adj=0, cex=1.2)

> text(2.3,3.0,“red: VS", col=“red",adj=0, cex=1.2)

Enhanced residual plots

> vaso.glm = glm(Response ~ log(Volume) + log(Rate), family=binomial, data=vaso.df)

> pear.r<-residuals(vaso.glm, type="pearson")

> dev.r<-residuals(vaso.glm, type="deviance")

> par(mfrow=c(1,2))

> plot(pear.r, ylab="residuals", main="Pearson",type="n")

> text(pear.r,cex=0.7)

> abline(h=0,lty=2)

> abline(h=2,lty=2,lwd=2)

> abline(h=-2,lty=2,lwd=2)

> plot(dev.r, ylab="residuals", main="Deviance",type="h")

> text(dev.r, cex=0.7)

> abline(h=0,lty=2)

> abline(h=2,lty=2,lwd=2)

> abline(h=-2,lty=2,lwd=2)

Diagnostics: Hat matrix diagonals

- Can define hat matrix diagonals (HMD’s) pretty much as in linear models
- HMD big if HMD > 3p/M

(M= no of covariate patterns)

- Draw index plot of HMD’s

Plotting HMD’s

> HMD<-hatvalues(vaso.glm)

> plot(HMD,ylab="HMD's",type="h")

> text(HMD,cex=0.7)

> abline(h=3*3/39, lty=2)

Hat matrix diagonals

- In ordinary regression, the hat matrix diagonals measure how “outlying” the covariates for an observation are
- In logistic regression, the HMD’s measure the same thing, but are down-weighted according to the estimated probability for the observation. The weights gets small if the probability is close to 0 or 1.
- In the vaso-constriction data, points 1,2,17 had very small weights, since the probabilities are close to 1 for these points.

Diagnostics: Cooks distance

- Can define an analogue of Cook’s distance for each point

CD = (Pearson resid )2 x HMD/(p*(1-HMD)2)

p = number of coeficients

- CD big if more than about 10% quantile of the chi-squared distribution on k+1 df, divided by k+1
- Calculate with qchisq(0.1,k+1)/(k+1)
- But not that reliable as a measure

Cooks D: calculating and plotting

p<-3

CD<-cooks.distance(vaso.glm)

plot(CD,ylab="Cook's D",type="h",

main="index plot of Cook's distances")

text(CD, cex=0.7)

bigcook<-qchisq(0.1,p)/p

abline(h=bigcook, lty=2)

Diagnostics: leave-one-out deviance change

- If the ith covariate pattern is left out, the change in the deviance is approximately

(Dev. Res) 2 + (Pearson. Res)2HMD/(1-HMD)

Big if more than about 4

Deviance change: calculating and plotting

> dev.r<-residuals(vaso.glm,type="deviance")

> Dev.change<-dev.r^2 + pear.r^2*HMD/(1-HMD)

> plot(Dev.change,ylab="Deviance change",

type="h")

> text(Dev.change, cex=0.7)

> bigdev<-4

> abline(h=bigdev, lty=2)

All together

influenceplots(vaso.glm)

Should we delete points?

- How influential are the 3 points?
- Can delete each in turn and examine changes in coefficients, predicted probabilities
- First, coefficients:

Deleting: None 31 4 18 All 3

(Intercept) -2.875 -3.041 -5.206 -4.758 -24.348

log(Volume) 5.179 4.966 8.468 7.671 39.142

log(Rate) 4.562 4.765 7.455 6.880 31.642

Should we delete points (2)?

- Next, fitted probabilities:
- Conclusion: points 4 and 18 have a big effect.

delete points

Fitted at None 31 4 18 4 and 18 All 3

point 31 0.722 0.627 0.743 0.707 0.996 0.996

point 4 0.075 0.073 0.010 0.015 0.000 0.000

point 18 0.106 0.100 0.018 0.026 0.000 0.000

Should we delete points (3)?

- Should we delete?
- They could be genuine – no real evidence they are wrong
- If we delete them, we increase the regression coefficients, make fitted probabilities more extreme
- Overstate the predictive ability of the model

Residuals for ungrouped data

- If all cases have distinct covariate patterns, then the residuals lie along two curves (corresponding to success and failure) and have little or no diagnostic value.
- Thus, there is a pattern even if everything is OK.

Formulas

- Pearson residuals: for ungrouped data, residual for i th case is

Formulas (cont)

- Deviance residuals: for ungrouped data, residual for i th case is

Use of plot function

plot(kyphosis.glm)

Analogue of R2?

- There is no satisfactory analogue of R2 for logistic regression.
- For the “small m big n” situation we can use the residual deviance, since we can obtain an approximate p-value.
- For other situations we can use the Hosmer –Lemeshow statistic (next slide)

Hosmer-Lemeshow statistic

- How can we judge goodness of fit for ungrouped data?
- Can use the Hosmer-Lemeshow statistic, which groups the data into cases having similar fitted probabilities
- Sort the cases in increasing order of fitted probabilities
- Divide into 10 (almost) equal groups
- Do a chi-square test to see if the number of successes in each group matches the estimated probability

Kyphosis data

Divide probs into 10 classes : lowest 10%, next 10%......

Class 1 Class 2 Class 3 Class 4 Class 5

Observed 0’s 9 8 8 7 8

Observed 1’s 0 0 0 1 0

Total obs 9 8 8 8 8

Expected 1’s 0.022 0.082 0.199 0.443 0.776

Class 6 Class 7 Class 8 Class 9 Class 10

Observed 0’s 8 5 5 3 3

Observed 1’s 0 3 3 5 5

Total obs 8 8 8 8 8

Expected 1’s 1.023 1.639 2.496 3.991 6.328

Note: Expected = Total.obs x average prob

In R, using the kyphosis data

Result of fitting model

> HLstat(kyphosis.glm)

Value of HL statistic = 6.498

P-value = 0.592

A p-value of less than 0.05 indicates problems. No problem indicated for the kyphosis data – logistic appears to fit OK.

The function HLstat is in the “330 functions”

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