Logistic Regression in STATA

1
Logistic Regression in STATA

• The logistic regression programs in STATA use
  maximum likelihood estimation to generate the
  logit (the logistic regression coefficient, which
  corresponds to the natural log of the OR for each
  one-unit increase in the level of the regressor
  variable).
• The resulting ORs are maximum-likelihood
  estimates (MLEs) of the uniform effect (OR)
  across strata of the model covariates. Thus they
  are pooled (uniform, common) estimates of the OR
  and in this sense are adjusted for all regressors
  included in the model.

2
STATA Logistic Regression Commands

• The logistic command in STATA yields odds
  ratios.
• . logistic sil hpv2 age
• Logit estimates
  Number of obs 595
• 
  LR chi2(2) 155.28
• 
  Prob gt chi2 0.0000
• Log likelihood -332.23774 Pseudo R2
  0.1894
• --------------------------------------------------
  ----------------------------
• sil Odds Ratio Std. Err. z
  Pgtz 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• hpv2 17.23939 5.10605 9.61
  0.000 9.647363 30.80598
• age .9947965 .0146842 -0.35
  0.724 .9664283 1.023997
• --------------------------------------------------
  ----------------------------

3
STATA Logistic Regression Commands

• The logit command in STATA yields the actual
  beta coefficients.
• logit sil hpv2 age
• Logit estimates
  Number of obs 595
• 
  LR chi2(2) 155.28
• 
  Prob gt chi2 0.0000
• Log likelihood -332.23774 Pseudo
  R2 0.1894
• --------------------------------------------------
  ----------------------------
• sil Coef. Std. Err. z
  Pgtz 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• hpv2 2.847197 .2961851 9.61
  0.000 2.266685 3.427709
• age -.0052171 .014761 -0.35
  0.724 -.0341482 .0237139
• _cons -.2873639 .4075825 -0.71
  0.481 -1.086211 .5114832
• --------------------------------------------------
  ----------------------------
• Note that 17.23939 exp2.847197 as in the
  previous slide

4
STATA Commands for Multilevel Categorical
Variables in Logistic Regression Models

• Categorized continuous variables should be
  entered in regression models as a series of
  indicator variables
• for each category a variable is created in which
  observations falling in that category are coded
  1" and all other observations are coded 0",
  thus the variable is represented in the model as
  a series of indicator terms, with the reference
  category left out of the model.
• Any categories of a variable that get left out of
  the model become part of the reference group
  (because those observations will be coded 0 for
  each indicator term left in the model).

5
STATA Commands for Multilevel Categorical
Variables in Logistic Regression Models

• If categorized continuous variables are entered
  in models as if they were continuous, that is, as
  one term rather than a series of indicator
  variables, the program will treat the values as a
  continuous distribution, with each observation in
  a category having the same value. The resulting
  odds ratio will correspond to each one unit
  increase in the category coding. This will not
  produce a meaningful result unless the coding can
  be interpreted as linear increments from one
  category to another.
• STATA has a convenient command that makes it
  unnecessary to create the indicator terms for
  multilevel categorical variables. The xi
  command creates a series of indicator variables
  for variables marked i.variablename by
  recognizing each value as a category. When used
  with the logistic or logit commands, STATA uses
  the lowest value as the reference category, which
  it drops out of the model. It is necessary to
  make sure that the variable coding reflects the
  desired categorization and reference level.

6

• xilogistic sil i.partner
• i.partner
• Logit estimates
  Number of obs 580
• 
  LR chi2(12) 51.48
• 
  Prob gt chi2 0.0000
• Log likelihood -374.7638
  Pseudo R2 0.0643
• --------------------------------------------------
  ----------------------------
• sil Odds Ratio Std. Err.
  z Pgtz 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• _Ipartner_2 2.008163 .5687573 2.46
  0.014 1.152708 3.498473
• _Ipartner_3 2.231293 .6428015 2.79
  0.005 1.26864 3.924413
• _Ipartner_4 2.40158 .7180631 2.93
  0.003 1.336568 4.315221
• _Ipartner_5 3.322094 1.050245 3.80
  0.000 1.787778 6.173199
• _Ipartner_6 2.693878 1.338111 2.00
  0.046 1.017575 7.13164
• _Ipartner_7 7.836735 6.346424 2.54
  0.011 1.602531 38.32338
• _Ipartner_9 13.71429 14.85759 2.42
  0.016 1.64063 114.6399
• _Ipartner_10 7.183673 2.981345 4.75
  0.000 3.184811 16.20353
• _Ipartner_12 5.877551 6.864814 1.52
  0.129 .5956855 57.99303

7
Note constant from logit model ln(odds of
reference category) logit ln(odds)-constant,
constant ln(odds of reference
value ORexp(logit)
8
Interpreting logistic regression model
coefficients for continuous variables

• When a logistic regression model contains a
  continuous independent variable, interpretation
  of the estimated coefficient depends on how it is
  entered into the model and the particular units
  of the variable
• To interpret the coefficient, we assume that the
  logit is linear in the variable
• The slope coefficient gives the change in the log
  odds for an increase of 1 unit in x.

9
Interpreting logistic regression model
coefficients for continuous variables

• Sometimes a unit increase may not be meaningful
  or considered important
• If we are interested in estimating the increased
  odds instead for every 5 year increase. We can
  use the formula
• OR (c)Exp(cß1)
• (95 CIexp(cß11.96cSEß1)

10
P-values for Trend

• The term trend generally refers to a monotonic,
  though not necessarily linear, association
  between increasing levels of exposure and the
  probability of the outcome.
• Although examining effect estimates and
  confidence intervals over levels of exposure is
  the most informative manner of evaluating a
  dose-response trend, it has been conventional to
  report p-values for the null hypothesis of no
  monotonic association between exposure and
  disease, that is, a test for trend.
• The p-value corresponding to the coefficient of a
  variable entered on an appropriate continuous
  scale in a regression model can be interpreted as
  a p-value for trend. For statistical hypothesis
  testing of trend in stratified analysis, see
  Rothman and Greenland.

11

• . logit sil lifepart
• Iteration 0 log likelihood -407.27031
• Iteration 1 log likelihood -398.61876
• Iteration 2 log likelihood -397.39561
• Iteration 3 log likelihood -397.33389
• Iteration 4 log likelihood -397.33374
• Logit estimates
  Number of obs 591
• 
  LR chi2(1) 19.87
• 
  Prob gt chi2 0.0000
• Log likelihood -397.33374
  Pseudo R2 0.0244
• --------------------------------------------------
  ----------------------------
• sil Coef. Std. Err. z
  Pgtz 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• lifepart .0493962 .0139827 3.53
  0.000 .0219906 .0768017
• _cons -.0989464 .1091532 -0.91
  0.365 -.3128828 .11499
• --------------------------------------------------
  ----------------------------

12
Interpretation

• For every increase in the number of partners, the
  risk or odds of SIL increases 5.
• If we are interested in estimating the increased
  odds instead for every 5 partners. We can use the
  formula
• OR (5)Exp(5 .0493962)3.43
• For every increase of 5 additional partners, the
  odds for SIL increases 3.43 times
• Be careful.. Validity of statements may be
  questionable since additional risk of SIL for 10
  partners may be quite different for 5 partners,
  but this is an unavoidable dilemma when
  continuous variables are modeled linearly in the
  logit.

13
Significance testing

• Does the model that includes the variable in
  question tell more about the outcome variable
  than a model that does not include that variable?
• In general you are comparing observed values of
  the response variable to the predicted values
  obtained from models with and without the
  variables in question
• Log likelihood ratio test is obtained by
  multiplying the difference between the two values
  by -2
• Follows a chi square distribution with the
  degrees of freedom the difference between the
  number of parameters in the 2 models

14
LR Test
. xilogistic sil i.hpv2 i.hpv2
_Ihpv2_0-1 (naturally coded _Ihpv2_0
omitted) Logit estimates
Number of obs 595
LR
chi2(1) 155.15
Prob gt chi2
0.0000 Log likelihood -332.30027
Pseudo R2 0.1893 -----------------
--------------------------------------------------
----------- sil Odds Ratio Std. Err.
z Pgtz 95 Conf. Interval ---------
-------------------------------------------------
------------------- _Ihpv2_1 17.30026
5.121828 9.63 0.000 9.683894
30.90687 -----------------------------------------
------------------------------------- .
lrtest,saving (0)
15
LR Test

• . xilogistic sil i.hpv2 income
• i.hpv2 _Ihpv2_0-1 (naturally
  coded _Ihpv2_0 omitted)
• Logit estimates
  Number of obs 592
• 
  LR chi2(2) 156.07
• 
  Prob gt chi2 0.0000
• Log likelihood -329.84288
  Pseudo R2 0.1913
• --------------------------------------------------
  ----------------------------
• sil Odds Ratio Std. Err. z
  Pgtz 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• _Ihpv2_1 17.3712 5.14748 9.63
  0.000 9.718504 31.04989
• income .9664517 .0766365 -0.43
  0.667 .8273373 1.128958
• --------------------------------------------------
  ----------------------------
• . lrtest, using (0)
• Warning observations differ 595 vs. 592
• Logistic likelihood-ratio test
  chi2(-1) -4.91
• 
  Prob gt chi2 .

16
LR Test adding income to the model

• -2(-332.30027 (-329.84288))2.457
• P08

17
Wald test

• Obtained by comparing the maximum likelihood
  estimate of the slope parameter, ß1, to an
  estimate of the standard error
• W ß1/SE (ß1)
• A Z test, the distribution is approximately
  standard normal
• Gives approximately equal answer to the LR test
  in large samples but may be different in smaller
  samples
• LR test seems to perform better in most situations

18
Variable selection

• Once we have obtained a model that contains
  essential variables, examine the variables in the
  model more closely
• Then check for interactions- For example an
  interaction between HPV status and race would
  indicate that the slope coefficient for HPV would
  be different by race/ethnicity

19
Interaction assessment

• Can take many forms
• When interaction is present, the association
  between the risk factor and the outcome variable
  differs, or depends in some way on the level of
  the covariate
• The covariate modifies the effect of the risk
  factor
• Consider a model with a dichotomous risk factor
  and the covariate, age. If the association
  between the covariate (age) and the outcome
  variable is the same within each level of the
  risk factor, then there is no interaction between
  the covariate and the risk factor
• Graphically, no interaction yields a model with
  parallel lines

20
Interaction assessment

• Need to be HWF
• Decide whether the effect of each variable varies
  importantly across race/ethnic categories.
• To make this decision, first note the magnitude
  of the difference between the ORs across strata.
• The tests of homogeneity have low power for
  detecting moderate effect-measure modification.
• The p-value on this test indicates whether there
  is adequate statistical power in the data to
  detect a difference, but a high p-value does not
  mean there is no effect-measure modification.
• In general, the significance level for
  heterogeneity worth exploring further should be
  set around 0.20-0.25

21
Observe stratum-specific estimates using
stratified analysis.

• cc sil hpv2, by (race)
• OR 95
  Conf. Interval M-H Weight
• -------------------------------------------------
  -----------------
• white 28.88889 8.69751
  148.2404 .8790698 (exact)
• african- 7.520408 2.891786
  21.71961 1.973154 (exact)
• hispanic 20.5 6.885033
  81.30018 1.073593 (exact)
• -------------------------------------------------
  -----------------
• Crude 17.30026 9.556989
  33.34827 (exact)
• M-H combined 15.85477 8.815859
  28.51382
• --------------------------------------------------
  -----------------
• Test of homogeneity (M-H) chi2(2) 3.80
  Prgtchi2 0.1494
• Test that combined OR 1
• Mantel-Haenszel chi2(1)
  123.18
• Prgtchi2 0.0000

22
Interaction assessment

• Use product terms in a multivariable logistic
  regression model in order to identify potential
  effect-measure modification (interaction) while
  adjusting for confounders.
• The p-value on the product term can be
  interpreted as a test of homogeneity.
• To model a product term for two continuous
  variables, a term must be created for the product
  of the two variables. The product term is entered
  into the model, along with each of the two
  variables.
• The xi command in STATA creates all of the
  required product terms for modeling interaction
  if at least one of the two variables is
  categorical. With this command the two variables
  do not have to be entered separately in the model
  because STATA does it for you.
• When entering a product term between a
  categorical and a continuous variable in a
  logistic regression model, we evaluate whether
  the entire dose-response of the continuous
  variable differs across strata of the categorized
  variable.

23

• . xilogistic sil i.hpv2 i.race
• Logit estimates
  Number of obs 595
• 
  LR chi2(3) 166.35
• 
  Prob gt chi2 0.0000
• Log likelihood -326.70089 Pseudo R2
  0.2029
• --------------------------------------------------
  ----------------------------
• sil Odds Ratio Std. Err. z
  Pgtz 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• _Ihpv2_1 15.86857 4.726791 9.28
  0.000 8.850939 28.45026
• _Irace_2 .5410994 .1348136 -2.47
  0.014 .3320491 .8817628
• _Irace_3 .4955735 .1097656 -3.17
  0.002 .3210507 .7649667
• --------------------------------------------------
  ----------------------------

24

• . xilogistic sil i.hpv2 i.race i.racehpv2
• Logit estimates
  Number of obs 595
• 
  LR chi2(5) 170.04
• 
  Prob gt chi2 0.0000
• Log likelihood -324.85603
  Pseudo R2 0.2074
• --------------------------------------------------
  ----------------------------
• sil Odds Ratio Std. Err.
  z Pgtz 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• _Ihpv2_1 28.88889 17.72576 5.48
  0.000 8.678548 96.16446
• _Irace_2 .6467662 .1711091 -1.65
  0.100 .3850814 1.086281
• _Irace_3 .515873 .1214782 -2.81
  0.005 .3251631 .8184354
• _IracXhpv2_2 .2603218 .1996824 -1.75
  0.079 .057888 1.170666
• _IracXhpv2_3 .7096154 .5830384 -0.42
  0.676 .1417926 3.551341
• --------------------------------------------------
  ----------------------------

25
Interaction assessment

• Use chunk test for entire collection of
  interaction terms
• Use LR test comparing main effects model with
  fuller model
• . lrtest, using (0)
• Logistic likelihood-ratio test chi2(-2)
  -3.69
• Prob
  gt chi2 0.15
• If the interaction term is not significant then
  drop the interaction term

26
Interaction Assessment

• If the interaction term is retained in the model,
  the estimated ORs for other variables confounded
  by race or another modifier should not be
  obtained from a model that enters race or
  another modifier in suboptimal form for the
  purpose of obtaining stratum-specific estimates.
• In an analysis that aims to estimate effects of
  several variables, we may use several different
  models to estimate the effects of interest. In
  this case, our goal is not the elaboration of a
  final model.

27
Confounding Assessment

• Confounding is used to describe a covariate that
  is associated both with the outcome variable of
  interest and the primary independent variable or
  risk factor of interest, but is not an
  intermediate variable in the causal pathway
• When both variables are present, the relationship
  is said to be confounded
• Only appropriate when there is no interaction
• Decisions regarding whether or not to adjust for
  potential confounding variables will depend on a
  combined assessment of prior knowledge, observed
  associations in the data, and sample size
  considerations

28
Confounding Assessment

• In practice we look for empirical evidence of
  confounding in data obtained from study
  populations, however, we must keep in mind that
  what we observe in such data may reflect
  selection and information bias affecting observed
  confounder-disease-exposure associations in a
  similar way to how these biases affect observed
  exposure-disease associations.
• Therefore, it is necessary to rely on prior
  knowledge of relevant associations in source
  populations. For example, if a variable is a
  known confounder but does not appear to be one in
  the data, this should create uncertainty
  regarding the validity of the data. Adjusting for
  this factor may not change the relative risk
  point estimate, but it may influence the standard
  error for this estimate, thus appropriately
  reflecting our uncertainty.
• If prior knowledge suggests a variable should not
  be a confounder but it appears to be one in the
  data, the confounding may have been introduced by
  the study methods (eg., as a result of matching
  in a case-control design). In this case it would
  be appropriate to adjust for this factor, if it
  is not an intervening (intermediate) variable.

29
Confounding Assessment

• When prior knowledge regarding exposure-covariate
  associations is insufficient and the number of
  covariates to consider is small, it may be
  desirable to adjust for all variables that appear
  to be important risk factors for the outcome, as
  long as they are not intervening variables.
• When a large number of potential confounders must
  be considered, the change-in-estimate variable
  selection strategy has been shown in simulation
  studies to produce more valid results for
  confounder detection than strategies that rely on
  p-values, unless the significance level for the
  p-value is raised to 0.2 or higher. There is more
  than one reasonable approach to variable
  selection in such situations the important thing
  is that the criterion for selection should be
  explicit and consistently applied.
• When examining continuous variables as potential
  confounders, careful assessment of the
  dose-response for the purpose of choosing the
  optimal scale or categorization should be carried
  out prior to the assessment of confounding.
  Inappropriate modeling of continuous variables
  may lead to incorrect decisions about whether or
  not to adjust for these variables and/or to
  inadequate control of confounding.

30
Confounding Assessment

• Begin with a model with the exposure-disease
  relationship For example if we are interested in
  the association between number of lifetime
  partners and SIL
• . xilogistic sil i.part1_4
• i.part1_4 _Ipart1_4_0-3 (naturally
  coded _Ipart1_4_0 omitted)
• Logit estimates
  Number of obs 593
• 
  LR chi2(3) 47.91
• 
  Prob gt chi2 0.0000
• Log likelihood -384.71
  Pseudo R2 0.0586
• --------------------------------------------------
  ----------------------------
• sil Odds Ratio Std. Err. z
  Pgtz 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• _Ipart1_4_1 2.113856 .5010233 3.16
  0.002 1.328387 3.363768
• _Ipart1_4_2 2.793651 .6974642 4.11
  0.000 1.712619 4.557047
• _Ipart1_4_3 5.120594 1.277563 6.55
  0.000 3.140146 8.350084
• -------

31
Confounding Assessment

• . . xilogistic sil i.curr_smk
• i.curr_smk _Icurr_smk_0-1 (naturally
  coded _Icurr_smk_0 omitted)
• Logit estimates
  Number of obs 595
• 
  LR chi2(1) 21.26
• 
  Prob gt chi2 0.0000
• Log likelihood -399.24786
  Pseudo R2 0.0259
• --------------------------------------------------
  ----------------------------
• sil Odds Ratio Std. Err. z
  Pgtz 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• _Icurr_smk_1 2.349306 .4454758 4.50
  0.000 1.620073 3.406784
• --------------------------------------------------
  ----------------------------
• .

32
Confounding Assessment

• . xilogistic sil i.curr_smk i.hpv2
• Logit estimates
  Number of obs 595
• 
  LR chi2(2) 161.58
• Prob gt
  chi2 0.0000
• Log likelihood -329.08559
  Pseudo R2 0.1971
• --------------------------------------------------
  ----------------------------
• sil Odds Ratio Std. Err.
  z Pgtz 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• _Icurr_smk_1 1.724126 .3709956 2.53
  0.011 1.130859 2.628631
• _Ihpv2_1 15.96255 4.749559 9.31
  0.000 8.909072 28.6004
• --------------------------------------------------
  ----------------------------

33
Confounding Assessment

• Examine the OR for the main risk factor to
  determine whether it is meaningfully different
  than the OR when controlling for confounding
• For previous examples the ORs are 2.34 vs. 1.72
• Can examine using the LR test
• May want to include the confounder if others
  would not trust the results if did not perform
  the adjustment
• May want to use a decision rule, depending on the
  subject matter (i.e., 10 change in the OR) and
  report the criterion
• Several have suggested a backward deletion
  strategy whereby you enter all main effects and
  confounders in the model and you eliminate
  one-by-one confounder which makes the smallest
  difference in the exposure-effect estimate
• Biases resulting from multiple confounders may
  cancel each other out or produce results that may
  not be easy to disentangle. Consider the
  following.

34
Model Checking

• When run diagnostics and why?
• This depends on what we want the model to do
• If our goal is to obtain approximately valid
  summary estimates of effect for a few key
  relationships, less rigorous checking is required
• If our goal is to predict outcomes for a given
  set of factors, more detailed checking is required

35
Model Checking

• Assuming the model contains those variables that
  should be in the model, we want to know how
  effectively the model we have describes the data
  (goodness of fit)
• A good-fitting model is not the same as a correct
  model
• Regression diagnostics can detect discrepancies
  between a model and data only within the range of
  the data and only if there are enough
  observations for adequate diagnostic power
• A model may appear to fit well in the central
  range of the data, but produce poor predictions
  for covariate values that are not well
  represented in the data

36
Model Checking

• Computation and evaluation of overall measures
• Examination of individual components of the
  summary statistics, often graphically
• Examination of other measures of the difference
  between components of the observed values versus
  the predicted values

37
Model Checking

• Check model results against results from
  stratified analysis
• Log-likelihood Ratio Test (also called Deviance
  Test)
• Tests of Regression
• Test the hypothesis that all the regression
  coefficients (except the intercept) are zero
• The R2 not recommended o use- may give a
  distorted impression

38
Model Checking

• Tests of fit (see RG, pp. 409-10 for test
  details)
• Tests for nonrandom incompatibilities between a
  model and data
• Compares the fit of an index model to a more
  elaborate reference model that contains it
• Small p-value suggests that the index model fits
  poorly relative to the reference model, that is,
  that the additional terms in the reference model
  improve the fit
• Large p-value does not mean that the index model
  fits well, only that the test did not detect an
  improvement in fit from the additional terms in
  the reference model
• Pearson residual (examining obs-pred/SE)
• Deviance residual
• Hosmer-Lemeshow goodness of fit statistic
• A grouping-based method based on values of the
  estimated probabilities
• Classification tables
• Table is the result of cross-classifying the
  outcome variable with a dichotomous variable
  whose values are derived from the estimated
  logistic probabilities

39
Multicolinearity

• Occurs when one or more independent variables can
  be approximately determined by some other
  variables in the model
• When there is multicolinearity, the estimated
  regression coefficients of the fitted model can
  be highly unreliable

40
Multiple testing

• Occurs from the many tests of significance that
  are typically carried out when selecting or
  eliminating variables from the model
• More likely to obtain statistical significance
  even is no real association exists

41
Influential observations

• Refers to data on individuals that may have a
  large influence on the estimated regression
  coefficients
• Methods assessing the possibility of influential
  observations should be considered