1 Logistic Regression EPP 245 Statistical Analysis of Laboratory Data.

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Logistic Regression

EPP 245

Statistical Analysis of

Laboratory Data

November 2, 2006 EPP 245 Statistical Analysis of Laboratory Data

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Generalized Linear Models

• The type of predictive model one uses depends on a number of issues; one is the type of response.

• Measured values such as quantity of a protein, age, weight usually can be handled in an ordinary linear regression model

• Patient survival, which may be censored, calls for a different method (next quarter)


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• If the response is binary, then can we use logistic regression models

• If the response is a count, we can use Poisson regression

• Other forms of response can generate other types of generalized linear models


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Generalized Linear Models

• We need a linear predictor of the same form as in linear regression βx

• In theory, such a linear predictor can generate any type of number as a prediction, positive, negative, or zero

• We choose a suitable distribution for the type of data we are predicting (normal for any number, gamma for positive numbers, binomial for binary responses, Poisson for counts)

• We create a link function which maps the mean of the distribution onto the set of all possible linear prediction results, which is the whole real line (-∞, ∞).

• The inverse of the link function takes the linear predictor to the actual prediction


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• Ordinary linear regression has identity link (no transformation by the link function) and uses the normal distribution

• If one is predicting an inherently positive quantity, one may want to use the log link since ex is always positive.

• An alternative to using a generalized linear model with an log link, is to transform the data using the log or maybe glog. This is a device that works well with measurement data but may not be usable in other cases


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∞0

Possible Means

-∞ ∞0

Predictors

Link= Log


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∞0

Possible Means

-∞ ∞0

Predictors

InverseLink= ex


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Logistic Regression

• Suppose we are trying to predict a binary variable (patient has ovarian cancer or not, patient is responding to therapy or not)

• We can describe this by a 0/1 variable in which the value 1 is used for one response (patient has ovarian cancer) and 0 for the other (patient does not have ovarian cancer

• We can then try to predict this response


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• For a given patient, a prediction can be thought of as a kind of probability that the patient does have ovarian cancer. As such, the prediction should be between 0 and 1. Thus ordinary linear regression is not suitable

• The logit transform takes a number which can be anything, positive or negative, and produces a number between 0 and 1. Thus the logit link is useful for binary data


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Possible Means

-∞ ∞0

Predictors

Link= Logit


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10

Possible Means

-∞ ∞0

Predictors

InverseLink= inverselogit


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1 1 1

logit log if 0 then logit( ) if 1 then logit( )1

logit if x then logit ( ) 0 if x then logit ( ) 11

1 1log log1

11 1

x

x

x x

x x

x x x

x x

pp p p p p

p

ex x x

e

e ee ee e ee e

1log log1

1

x

xx

x

ee e x

e


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Analyzing Tabular Data with Logistic Regression

• Response is hypertensive y/n

• Predictors are smoking (y/n), obesity (y/n), snoring (y/n) [coded as 0/1 for Stata, R does not care]

• How well can these 3 factors explain/predict the presence of hypertension?

• Which are important?


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input smoking obesity snoring hyp fw0 0 0 1 51 0 0 1 20 1 0 1 11 1 0 1 00 0 1 1 351 0 1 1 130 1 1 1 151 1 1 1 80 0 0 0 551 0 0 0 150 1 0 0 71 1 0 0 20 0 1 0 1521 0 1 0 720 1 1 0 361 1 1 0 15end


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. do hypertension-in

. input smoking obesity snoring hyp fw

smoking obesity snoring hyp fw 1. 0 0 0 1 5 2. 1 0 0 1 2 3. 0 1 0 1 1 4. 1 1 0 1 0 5. 0 0 1 1 35 6. 1 0 1 1 13 7. 0 1 1 1 15 8. 1 1 1 1 8 9. 0 0 0 0 55 10. 1 0 0 0 15 11. 0 1 0 0 7 12. 1 1 0 0 2 13. 0 0 1 0 152 14. 1 0 1 0 72 15. 0 1 1 0 36 16. 1 1 1 0 15 17. end

. end of do-file


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. logistic hyp smoking obesity snoring [fweight=fw]

Logistic regression Number of obs = 433 LR chi2(3) = 12.51 Prob > chi2 = 0.0058Log likelihood = -199.4582 Pseudo R2 = 0.0304

------------------------------------------------------------------------------ hyp | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- smoking | .9344708 .2598989 -0.24 0.807 .5417838 1.611779 obesity | 2.00433 .5714045 2.44 0.015 1.146316 3.504564 snoring | 2.391544 .950815 2.19 0.028 1.097143 5.213072------------------------------------------------------------------------------

. logit


------------------------------------------------------------------------------ hyp | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- smoking | -.0677749 .2781242 -0.24 0.807 -.6128882 .4773385 obesity | .6953096 .2850851 2.44 0.015 .136553 1.254066 snoring | .8719393 .3975737 2.19 0.028 .0927093 1.651169 _cons | -2.377661 .3801845 -6.25 0.000 -3.122809 -1.632513------------------------------------------------------------------------------


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Juul's IGF data

Description:

The 'juul' data frame has 1339 rows and 6 columns. It contains a reference sample of the distribution of insulin-like growth factor (IGF-I), one observation per subject in various ages with the bulk of the data collected in connection with school physical examinations.

Variables:

age a numeric vector (years). menarche a numeric vector. Has menarche occurred (code 1: no, 2: yes)? sex a numeric vector (1: boy, 2: girl). igf1 a numeric vector. Insulin-like growth factor ($mu$g/l). tanner a numeric vector. Codes 1-5: Stages of puberty a.m. Tanner. testvol a numeric vector. Testicular volume (ml).


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. clear

. insheet using "C:\TD\CLASS\K30-2006\juul.csv"

. summarize age menarch sex igf1 tanner testvol

Variable | Obs Mean Std. Dev. Min Max-------------+-------------------------------------------------------- age | 1334 15.09535 11.25288 .17 83 menarche | 704 1.475852 .4997716 1 2 sex | 1334 1.534483 .4989966 1 2 igf1 | 1018 340.168 171.0356 25 915 tanner | 1099 2.639672 1.76314 1 5-------------+-------------------------------------------------------- testvol | 480 7.895833 8.212571 1 30

. keep if age > 8(237 observations deleted)

. keep if age < 20(153 observations deleted)


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. summarize age menarch sex igf1 tanner testvol

Variable | Obs Mean Std. Dev. Min Max-------------+-------------------------------------------------------- age | 949 13.36911 3.238077 8.01 19.87 menarche | 519 1.506744 .5004369 1 2 sex | 949 1.553214 .4974224 1 2 igf1 | 737 397.4627 161.1272 71 915 tanner | 863 3.01854 1.740637 1 5-------------+-------------------------------------------------------- testvol | 401 9.164589 8.331137 1 30

. keep if menarche < 100(430 observations deleted). summarize age menarch sex igf1 tanner testvol

Variable | Obs Mean Std. Dev. Min Max-------------+-------------------------------------------------------- age | 519 13.43778 3.227661 8.03 19.75 menarche | 519 1.506744 .5004369 1 2 sex | 519 2 0 2 2 igf1 | 411 414.0803 160.9518 95 914 tanner | 436 3.307339 1.730601 1 5-------------+-------------------------------------------------------- testvol | 0

.


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. generate men1 = menarche -1

. logistic men1 age


------------------------------------------------------------------------------ men1 | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- age | 4.559845 .7038612 9.83 0.000 3.369442 6.170811------------------------------------------------------------------------------

. logistic men1 age tanner


------------------------------------------------------------------------------ men1 | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- age | 2.266544 .4811877 3.85 0.000 1.495044 3.436168 tanner | 5.616052 1.760354 5.51 0.000 3.038236 10.38104------------------------------------------------------------------------------

.


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. generate tan1 = tanner == 1



. generate tan4 = tanner ==4


. logistic men1 age tan2 tan3 tan4 tan5


------------------------------------------------------------------------------ men1 | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- age | 3.944062 .7162327 7.56 0.000 2.762915 5.630151 tan2 | .0444044 .0486937 -2.84 0.005 .0051761 .3809341 tan3 | .1369598 .095596 -2.85 0.004 .0348712 .5379227 tan4 | .6969611 .3898228 -0.65 0.519 .2328715 2.085935 tan5 | 9.169558 7.638664 2.66 0.008 1.791671 46.9287------------------------------------------------------------------------------

.


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Class prediction from expression arrays

• One common use of omics data is to try to develop predictions for classes of patients, such as – cancer/normal– type of tumor– grading or staging of tumors– many other disease/healthy or diagnosis of

disease type


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Two-class prediction

• Linear regression• Logistic regression• Linear or quadratic discriminant analysis• Partial least squares• Fuzzy neural nets estimated by genetic

algorithms and other buzzwords• Many such methods require fewer

variables than cases, so dimension reduction is needed


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Dimension Reduction

• Suppose we have 20,000 variables and wish to predict whether a patient has ovarian cancer or not and suppose we have 50 cases and 50 controls

• We can only use a number of predictors much smaller than 50

• How do we do this?


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• Two distinct ways are selection of genes and selection of “supergenes” as linear combinations

• We can choose the genes with the most significant t-tests or other individual gene criteria

• We can use forward stepwise logistic regression, which adds the most significant gene, then the most significant addition, and so on, or other ways of picking the best subset of genes


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Supergenes are linear combinations of genes. If g1, g2, g3, …, gp are the expression measurements for the p genes in an array, and a1, a2, a3, …, ap are a set of coefficients then g1 a1+ g2 a2+ g3 a3+ …+ gp ap is a supergene. Methods for construction of supergenes include PCA and PLS


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Choosing Subsets of Supergenes• Suppose we have 50 cases and 50

controls and an array of 20,000 gene expression values for each of the 100 observations

• In general, any arbitrary set of 100 genes will be able to predict perfectly in the data if a logistic regression is fit to the 100 genes

• Most of these will predict poorly in future samples


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• This is a mathematical fact

• A statistical fact is that even if there is no association at all between any gene and the disease, often a few genes will produce apparently excellent results, that will not generalize at all

• We must somehow account for this, and cross validation is the usual way


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Consequences of many variables

• If there is no effect of any variable on the classification, it is still the case that the number of cases correctly classified increases in the sample that was used to derive the classifier as the number of variables increases

• But the statistical significance is usually not there


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• If the variables used are selected from many, the apparent statistical significance and the apparent success in classification is greatly inflated, causing end-stage delusionary behavior in the investigator

• This problem can be improved using cross validation or other resampling methods


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Overfitting

• When we fit a statistical model to data, we adjust the parameters so that the fit is as good as possible and the errors are as small as possible

• Once we have done so, the model may fit well, but we don’t have an unbiased estimate of how well it fits if we use the same data to assess as to fit


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Training and Test Data

• One way to approach this problem is to fit the model on one dataset (say half the data) and assess the fit on another

• This avoids bias but is inefficient, since we can only use perhaps half the data for fitting

• We can get more by doing this twice in which each half serves as the training set once and the test set once

• This is two-fold cross validation


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• It may be more efficient to use 5- 10-, or 20-fold cross validation depending on the size of the data set

• Leave-out-one cross validation is also popular, especially with small data sets

• With 10-fold CV, one can divide the set into 10 parts, pick random subsets of size 1/10, or repeatedly divide the data


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Stepwise Logistic Regression

• Another way to select variables is stepwise

• This can be better than individual variable selection, which may choose many highly correlated predictors that are redundent

• A generic function stepwise can be used for many kinds of predictor functions in stata

1 Logistic Regression EPP 245 Statistical Analysis of Laboratory Data.

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