Logistic Regression

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Logistic Regression

• EPP 245
• Statistical Analysis of
• Laboratory Data

2
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)

3

• 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

4
Generalized Linear Models

• We need a linear predictor of the same form as in
  linear regression ßx
• 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 (-8, 8).
• 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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Possible Means
0
8
Link Log
0
8
-8
Predictors
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Possible Means
0
8
Inverse Link ex
0
8
-8
Predictors
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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
1
Link Logit
0
8
-8
Predictors
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Possible Means
0
1
Inverse Link inverse logit
0
8
-8
Predictors
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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
fw 0 0 0 1 5 1 0 0 1 2 0 1 0 1 1 1 1 0 1 0 0 0 1 1
35 1 0 1 1 13 0 1 1 1 15 1 1 1 1 8 0 0 0 0 55 1 0
0 0 15 0 1 0 0 7 1 1 0 0 2 0 0 1 0 152 1 0 1 0 72
0 1 1 0 36 1 1 1 0 15 end
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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
fweightfw Logistic regression
Number of obs 433
LR
chi2(3) 12.51
Prob gt chi2
0.0058 Log likelihood -199.4582
Pseudo R2 0.0304 -------------
--------------------------------------------------
--------------- hyp Odds Ratio Std.
Err. z Pgtz 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 Logistic regression
Number of obs
433
LR chi2(3) 12.51
Prob gt chi2
0.0058 Log likelihood -199.4582
Pseudo R2
0.0304 ------------------------------------------
------------------------------------ hyp
Coef. Std. Err. z Pgtz 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 (mug/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 gt 8 (237
observations deleted) . keep if age lt 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 lt 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 Logistic regression
Number of obs 519
LR chi2(1)
518.73
Prob gt chi2 0.0000 Log
likelihood -100.33214
Pseudo R2 0.7211 --------------------
--------------------------------------------------
-------- men1 Odds Ratio Std. Err.
z Pgtz 95 Conf. Interval ------------
-------------------------------------------------
---------------- age 4.559845
.7038612 9.83 0.000 3.369442
6.170811 -----------------------------------------
------------------------------------- . logistic
men1 age tanner Logistic regression
Number of obs 436
LR
chi2(2) 493.02
Prob gt chi2
0.0000 Log likelihood -55.587542
Pseudo R2 0.8160 -------------
--------------------------------------------------
--------------- men1 Odds Ratio Std.
Err. z Pgtz 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 tan2
tanner 2 . generate tan3 tanner 3 .
generate tan4 tanner 4 . generate tan5
tanner 5 . logistic men1 age tan2 tan3 tan4
tan5 Logistic regression
Number of obs 519
LR chi2(5)
568.74
Prob gt chi2 0.0000 Log
likelihood -75.327218
Pseudo R2 0.7906 --------------------
--------------------------------------------------
-------- men1 Odds Ratio Std. Err.
z Pgtz 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 dont 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