Lecture 16:Logistic Regression:

Goodness of FitInformation CriteriaROC analysis

BMTRY 701Biostatistical Methods II

Goodness of Fit

A test of how well the model explains the data Applies to linear models and generalized linear

models How to do it? It is simply a comparison of the “current” model

to a perfect model• What would the estimated likelihood function be in a

perfect model?• What would the estimated log-likelihood function be in

a perfect model

Set up as a hypothesis test

Ho: current model H1: perfect model

Recall the G2 statistic comparing models:

G2 = Dev(0) - Dev(1) How many parameters are there in the null

model? How many parameters are there in the perfect

model?

Goodness of Fit test

Perfect model: Assumed to be ‘saturated’ in most cases

That is, there is a parameter for each combination of predictors

In our model? that is likely to be close to N due to the number of continuous variables

Define c = number of parameters in saturated model

Deviance goodness of fit: Dev(0)

Goodness of Fit test

Deviance goodness of fit: Dev(0)

If Dev(Ho) < χ2(c-p),1-α, conclude H0

If Dev(Ho) > χ2(c-p),1-α conclude H1

Why arent we subtracting deviances?

GoF test for Prostate Cancer Model

> mreg1 <- glm(cap.inv ~ gleason + log(psa) + vol + factor(dpros),+ family=binomial)> mreg0 <- glm(cap.inv ~ gleason + log(psa) + vol, family=binomial)> mreg1Coefficients: (Intercept) gleason log(psa) vol -8.31383 0.93147 0.53422 -0.01507 factor(dpros)2 factor(dpros)3 factor(dpros)4 0.76840 1.55109 1.44743

Degrees of Freedom: 378 Total (i.e. Null); 372 Residual (1 observation deleted due to missingness)Null Deviance: 511.3 Residual Deviance: 377.1 AIC: 391.1

Test Statistic: 377.1 ~ χ2(380 - 7)

Threshold: χ2(373),1-α, = 419.0339

p-value = 0.43

More Goodness of Fit

There are a lot of options! Deviance GoF is just one

• Pearson Chi-square • Hosmer-Lemeshow• etc

Principles, however, are essentially the same

GoF is not that commonly seen in medical research because it is rarely very important

Information Criteria

Information criterion is a measure of the goodness of fit of an estimated statistical model.

It is grounded in the concept of entropy, • offers a relative measure of the information lost • describes the tradeoff precision and complexity of the model.

An IC is not a test on the model in the sense of hypothesis testing

it is a tool for model selection. Given a data set, several competing models may be

ranked according to their IC The model with the lowest IC is chosen as the “best”

Information Criteria

IC rewards goodness of fit, but also includes a penalty that is an increasing function of the number of estimated parameters.

This penalty discourages overfitting. The IC methodology attempts to find the model that best

explains the data with a minimum of free parameters. More traditional approaches such as LRT start from a

null hypothesis. IC judges a model by how close its fitted values tend to

be to the true values. the AIC value assigned to a model is only meant to rank

competing models and tell you which is the best among the given alternatives.

Akaike Information Criteria (AIC)

pLikAIC 2log2 Akaike, Hirotugu (1974). "A new look at the statistical model identification". IEEE Transactions on Automatic Control 19 (6): 716–723..

Bayesian Information Criteria

)ln(log2 NpLikBIC

Schwarz, Gideon E. (1978). "Estimating the dimension of a model". Annals of Statistics 6 (2): 461–464.

AIC versus BIC

BIC and AIC are similar Different penalty for number of parameters The BIC penalizes free parameters more

strongly than does the AIC. Implications: BIC tends to choose smaller

models The larger the N, the more likely that AIC and

BIC will disagree on model selection

)ln( . 2 Npvsp

Prostate cancer models

We looked at different forms for volume:

A: volume as continuous

B: volume as binary (detectable vs. undetectable)

C: 4 categories of volume

D: 3 categories of volume

E: linear + squared term for volume

AIC vs. BIC (N=380)

p -2logLik AIC BIC

A: continuous 8 376.0 392.0 423.5

B: binary 8 375.2 391.2 422.7

C: 4 categories 10 373.6 393.6 433.0

D: 3 categories 9 375.2 393.2 428.6

E: quadratic 9 376.0 394.0 429.4

AIC vs. BIC if N is multiplied by 10 (N=3800)

p -2logLik AIC BIC

A: continuous 8 3760.0 3776.0 3825.9

B: binary 8 3752.0 3768.0 3817.9

C: 4 categories 10 3736.0 3756.0 3818.4

D: 3 categories 9 3751.9 3769.9 3826.1

E: quadratic 9 3760.0 3778.0 3834.2

ROC curve analysis

Receiver Operating Characteristic Curve Analysis

Traditionally, looks at the sensitivity and specificity of a ‘model’ for predicting an outcome

Question: based on our model, can we accurately predict if a prostate cancer patient has capsular penetration?

ROC curve analysis

Associations between predictors and outcomes is not enough

Need ‘stronger’ relationship Classic interpretation of sens and spec

• a binary test and a binary outcome• sensitivity = P(test + | true disease)• specificity = P(test - |true no disease)

What is test + in our dataset? What does the model provide for us?

ROC curve analysis

0.0

00

.25

0.5

00

.75

1.0

0S

ens

itivi

ty/S

pec

ifici

ty

0.00 0.25 0.50 0.75 1.00Probability cutoff

Sensitivity Specificity

Fitted probabilities

The fitted probabilities are the probability that a NEW patient with the same ‘covariate profile’ will be a “case” (e.g., capsular penetration, disease, etc.)

We select a probability ‘threshold’ to determine whether a patient is defined as a case or not

Some options:• high sensitivity (e.g., cancer screens)• high specificity (e.g., PPD skin test for TB)• maximize the sum of sens and spec

ROC curve

. xi: logit capsule i.dpros detected gleason logpsai.dpros _Idpros_1-4 (naturally coded; _Idpros_1 omitted)

Iteration 0: log likelihood = -255.62831Iteration 1: log likelihood = -193.51543Iteration 2: log likelihood = -188.23598Iteration 3: log likelihood = -188.04747Iteration 4: log likelihood = -188.0471

Logistic regression Number of obs = 379 LR chi2(6) = 135.16 Prob > chi2 = 0.0000Log likelihood = -188.0471 Pseudo R2 = 0.2644

------------------------------------------------------------------------------ capsule | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- _Idpros_2 | .7801903 .3573241 2.18 0.029 .079848 1.480533 _Idpros_3 | 1.606646 .3744828 4.29 0.000 .8726729 2.340618 _Idpros_4 | 1.504732 .4495287 3.35 0.001 .6236723 2.385793 detected | -.5719155 .2570359 -2.23 0.026 -1.075697 -.0681344 gleason | .9418179 .1648245 5.71 0.000 .6187677 1.264868 logpsa | .5152153 .1547649 3.33 0.001 .2118817 .8185488 _cons | -8.275811 1.056036 -7.84 0.000 -10.3456 -6.206018------------------------------------------------------------------------------

ROC curve

0.0

00

.25

0.5

00

.75

1.0

0S

ens

itivi

ty

0.00 0.25 0.50 0.75 1.001 - Specificity

Area under ROC curve = 0.8295

How to interpret?

Every point represents a patient(s) in the dataset Question: if we use that person’s fitted

probability as the threshold, what are the sens and spec values?

Empirically driven based on the fitted probabilities

Choosing the threshold:• high sens or spec• maximize both? the point on ROC curve closest to

the upper left corner

AUC of ROC curve

AUC = Area Under the Curve 0.5 < AUC < 1 AUC = 1 if the model is perfect AUC = 0.50 if the model is no better than chance “Good” AUC?

• context specific• for some outcomes, there are already good diagnostic

measures so AUC would need to be very high• for others, if there is very little, even an AUC of 0.70

would be useful.

Utility in model selection

If the goal of the modeling is prediction, AUC can be used to determine the ‘best’ model

A variable may be associated with the outcome, but not add much in terms of prediction

Example:• Model 1: gleason + logPSA + detectable + dpros• Model 2: gleason + logPSA + detectable• Model 3: gleason + logPSA

ROC curve of models 1, 2, and 3

False positive rate

Tru

e p

osi

tive

ra

te

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

1: AUC=0.832: AUC=0.803: AUC=0.79

Sensitivity and Specificity

For ‘true’ use, you need to choose a cutoff. The AUC of the ROC curve tells you about

prediction of model But, not directly translatable into ‘accuracy’ of a

given threshold

phat = 0.50 cutoff

Logistic model for capsule

-------- True --------Classified | D ~D | Total-----------+--------------------------+----------- + | 100 39 | 139 - | 53 187 | 240-----------+--------------------------+----------- Total | 153 226 | 379

Classified + if predicted Pr(D) >= .5True D defined as capsule != 0--------------------------------------------------Sensitivity Pr( +| D) 65.36%Specificity Pr( -|~D) 82.74%Positive predictive value Pr( D| +) 71.94%Negative predictive value Pr(~D| -) 77.92%--------------------------------------------------False + rate for true ~D Pr( +|~D) 17.26%False - rate for true D Pr( -| D) 34.64%False + rate for classified + Pr(~D| +) 28.06%False - rate for classified - Pr( D| -) 22.08%--------------------------------------------------Correctly classified 75.73%--------------------------------------------------

phat = 0.25 cutoff

Logistic model for capsule

-------- True --------Classified | D ~D | Total-----------+--------------------------+----------- + | 137 96 | 233 - | 16 130 | 146-----------+--------------------------+----------- Total | 153 226 | 379

Classified + if predicted Pr(D) >= .25True D defined as capsule != 0--------------------------------------------------Sensitivity Pr( +| D) 89.54%Specificity Pr( -|~D) 57.52%Positive predictive value Pr( D| +) 58.80%Negative predictive value Pr(~D| -) 89.04%--------------------------------------------------False + rate for true ~D Pr( +|~D) 42.48%False - rate for true D Pr( -| D) 10.46%False + rate for classified + Pr(~D| +) 41.20%False - rate for classified - Pr( D| -) 10.96%--------------------------------------------------Correctly classified 70.45%--------------------------------------------------