STA 107: Logistic Regression and Categorical Data Analysis

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STA 107 Logistic Regression and Categorical
Data Analysis

• Lecturer Dr. Daisy Dai
• Department of Medical Research

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Contents

• Binary Logit Analysis
• Simple Logistic Regression
• Multiple Logistic Regression
• Stepwise or Backward Model Selections
• Collinearity

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Categorical Data Analysis

• Binomial Test
• Chi-square Test
• Fishers Exact Test
• McNemars Test
• Cochran-Mantel-Haenszel Test

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Binomial test

• Make inference about a proportion of binary
  outcomes by comparing the confidence interval of
  a proportion to target.

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Case Study Genital Wart

• A company markets a therapeutic product for
  genital warts with a known cure rate of 40 in
  the general population. In a study of 25
  patients with genital warts treated with this
  product, patients were also given high doses of
  vitamin C. As shown in Table on the next page,
  14 patients were cured. Is this consistent with
  the cure rate in the general population?

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Treatment to Genital Wart
ID Effectiveness
1 YES
2 NO
3 YES
4 NO
5 YES
6 YES
7 NO
8 YES
9 NO
10 NO
11 YES
12 NO
13 YES
14 NO
ID Effectiveness
15 YES
16 NO
17 NO
18 YES
19 YES
20 NO
21 YES
22 YES
23 NO
24 YES
25 YES
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Results

• 64 (16/25) of patient were cured by the
  treatment.
• The 95 confidence interval extends from 44 to
  80
• If the probability of "success" in each trial or
  subject is 0.300, then the chance of observing 16
  or more successes in 25 trials is 0.045
  (p-value).
• The cure rate of genital wart by the experimental
  therapy was significantly higher than 30.

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Fishers Exact Test

• A conservative non-parametric test about a
  relationship between two categorical variables.
  The groups in comparison should be independent.

Responders Non-responders Total
Group 1 N11 N12 N11N12
Group 2 N21 N22 N21N22
Combined N11N21 N12 N22 N
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Case Study CHF Incidence

• A new adenosine-releasing agent (ARA), thought
  to reduce side effects in patients undergoing
  coronary artery bypass surgery (CABG), was
  studied in a pilot trial.

CHF No CHF Total
ARA 2 (6) 33 35
Placebo 5 (25) 20 25
Combined 7 53 60
Fishers exact test p0.0455
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Chi-square test

• Test a relationship between two categorical
  variables. Groups should be independent. The
  chi-square test assumes that the expected value
  for each cell is five or higher.

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Case Study ADR Frequency with Antibiotic
Treatment

• A study was conducted to monitor the incidence
  of GI adverse drug reactions of a new antibiotic
  used in lower respiratory tract infections.

Responders Non-responders Total
Test (new antibiotic) 22 (33) 44 66
Control (erythromycin) 28 (54) 24 53
Combined 50 (42) 68 118
Chi-square test p0.0252 Fishers exact test
p0.0385
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McNemars test

• Compare response rates in binary data between
  two related populations. Its analogous to
  Chi-square test or Fishers exact test for
  independent populations.

After Before Responders Non-responders
Responders A B
Non-responder C D
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Case Study Bilirubin

• A study was conduct to evaluate the toxicity
  side effect of an experimental therapy. Patients
  (n86) were treated with the experimental drug
  for 3 months. Clinical lab measured bilirubin
  levels of each patient at baseline and 3 months
  after therapy.

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Results of McNemars Test
After Before Normal Abnormally high
Normal 60 14
Abnormally high 6 6

• At baseline, 14 (12/86) of patients had
  abnormally high bilirubin level.
• At 3 months post treatment, 23 (20/86) of
  patients had abnormally high bilirubin level.
• P-value 0.1175
• Odds ratio 2.3 95 CI 0.8 - 7.4
• There is no enough evidence to prove the
  increasing risk of high bilirubin due to
  treatment.

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Cochran-Mantel-Haenszel (CMH) Test

• The Cochran-Mantel-Haenszel test is a method to
  compare the probability of an event among
  independent groups in stratified samples.
• The stratification factor can be study center,
  gender, race, age groups, obesity status or
  disease severity. These underlying
  sub-population can be confounding factors that
  affect the associations between risk factors and
  the outcome variables.

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Case Study Diabetic Ulcers

• A multi-center study with 4 centers is testing an
  experimental treatment, Dermotel, used to
  accelerate the healing of dermal foot ulcers in
  diabetic patients. Sodium hyaluronate was used
  in a control group. Patients who showed a
  decrease in ulcer size after 20 weeks treatment
  of at least 90 surface area measurements were
  considered responders. The numbers of
  responders in each group are shown in Table 19.2
  for each study center. Is there an overall
  difference in response rates between the Dermotel
  and control groups?

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Response Frequencies by Study Center
Study Center Treatment Group Response Non-Response
1 Dermotel 26 (87) 4
1 Control 18 (62) 11
2 Dermotel 8 (73) 3
2 Control 7 (58) 5
3 Dermotel 7 (58) 5
3 Control 4 (40) 6
4 Dermotel 11 (65) 6
4 Control 9 (64) 5
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• The interest in this study is to compare the
  response rate between two treatment. Because the
  study was conducted in four centers, it is
  concerned that some potential influences of study
  center on the response rate. By including the
  study center, the researcher can examine
  associations between the treatment and the
  response rate while adjusting (controlling) for
  the effect of study center.
• Cochran-Mantel-Haenszel Test assumes a common
  odds ratio and test the null hypothesis that the
  explanatory variable X (treatment) and the
  outcome variable Y (response rate) are
  conditionally independent, given the control
  variable Z (study center). In other words, CMH
  tests whether the response is conditionally
  independent of the explanatory variable when
  adjusting for the control variable.  
• One can also measures average conditional
  association between the explanatory (treatment)
  and the response variable by calculating the
  common odds ratio conditioned on the control
  variable (study center).

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Results
Response Rates Response Rates
Study Center Active (n) Control (n) Chi-Square p-Value
1 86.7 (30) 62.1 (29) 4.706 0.030
2 72.7 (11) 58.3 (12) 0.524 0.469
3 58.3 (12) 40.0 (10) 0.733 0.392
4 64.7 (17) 64.3 (14) 0.001 0.981
Overall 74.3 (70) 58.5 (65) 40.39 0.044
P-value from CMH test
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Chi-Square Test, Ignoring Strata
Group Response Non-Response Total
Active Control 52 (74.3) 38 (58.5) 18 27 70 65
Total 90 (100.0) 45 135
Chi-square value 3.798, p 0.051
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Counter-Intuitive Combined Results
Stratum Group Responders Non-Responders Total Response Rate
1 A 10 38 48 21
B 4 21 25 16
2 A 20 10 30 67
B 27 17 44 61
Combined A 30 48 78 38
B 31 38 69 45
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Logistic Regression
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Logistic Regression

• Logistic Regression are methods to identify the
  associations between a categorical outcome
  variable and explanatory variables.
• In most cases, the outcome variable is
  dichotomous. The explanatory variables can be
  categorical or continuous. The probability of the
  outcome variable can be predicted by the values
  of explanatory variables.
• Dichotomous outcome variable ? explanatory
  variables
• Log(P/(1-P))a b1 x1 b2 x2 b3 x3 b4
  x4

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Odds Ratio

• Let Y be the dichotomous variable where y1
  indicates an event and y0 indicates no events
• Oddprobability of an event/probability of no
  event
• P(Y1)/P(Y0)P(Y1)/(1-P(Y0))
• Odds RatioOdds in the Test Group/Odd in the
  Control Group
• Logistic Model Log(Odds Ratio of an event) ?
  explanatory variables
• Log (odds ratio)a b1 x1 b2 x2 b3 x3
  b4 x4

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Case Study CHF Incidence

• Odd of CHF incidence in the ARA
  group(2/35)/(33/35)2/336.
• Odd of CHF incidence in the Placebo
  group(5/25)/(20/25)20.
• Odds RatioOdd in the ARA group/odd in the
  Placebo group(2/33)/(5/20)0.24
• The risk (odd) of CHF incidence in the ARA group
  is only 24 the risk (odd) in the Placebo group.

• A new adenosine-releasing agent (ARA), thought
  to reduce side effects in patients undergoing
  coronary artery bypass surgery (CABG), was
  studied in a pilot trial.

CHF No CHF Total
ARA 2 (5.7) 33 35
Placebo 5 (25) 20 25
Combined 7 53 60
Fishers exact test p0.0455
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Properties of Odds Ratio

• Odds ratio is non-negative.
• If odds ratiolt1, then the risk is smaller than
  control.
• If odds ratiogt1, then the risk is larger than
  control.
• Odds ratio of no event1/odds ratio of an event.
• One can calculate the confidence interval of an
  odds ratio. The confidence interval of a
  significance odds ratio does not contain 1.

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Case Study CHF Incidence

• odds Ratio for ARA versus Control(2/33)/(5/20)0.
  24lt1. So the risk of CHF incidence in the ARA
  group is relatively smaller.
• One can also calculate odds ratio for Control
  versus ARA as 1/0.244.1gt1, which indicates the
  risk (odd) of CHF in Placebo group is 4.1 fold of
  risk in ARA group.

• A new adenosine-releasing agent (ARA), thought
  to reduce side effects in patients undergoing
  coronary artery bypass surgery (CABG), was
  studied in a pilot trial.

CHF No CHF Total
ARA 2 (5.7) 33 35
Placebo 5 (25) 20 25
Combined 7 53 60
Fishers exact test p0.0455
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Logistic Probability Curve

• Log(p/(1-p))abx
• p/1-pexp(abx)
• p1/(1exp(-a-bx))

Probability
X
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Logistic Regression vs. Linear Regression
Common In regression we are looking for a
dependence of one variable, the dependent
variable, on other, the independent variable(s).

• In linear regression the dependent variable is
  continuous
• The relationship is summarized by a regression
  equation consisting of a slope and an intercept.
  In increases with unit increase in the
  independent variable, and the intercept
  represents the value of the dependent variable
  when the independent variable takes the value
  zero.

• in logistic regression the dependent variable is
  binary.
• In logistic regression the slope represents the
  change in log odds for a unit increase in the
  independent variable and the regression we are
  interested in the simultaneous relationship
  between one dependent variable and a number of
  independent variables.

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Case Study Relapse Rate in AML

• One hundred and two patients with acute
  myelogenous leukemia (AML) in remission were
  enrolled in a study of a new antisense
  oligonucleotide (asODN). The patients were
  randomly assigned to receive a 10-day infusion of
  asODN or no treatment (Control), and the effects
  were followed for 90 days. The time of remission
  from diagnosis or prior relapse (X, in months) at
  study enrollment was considered an important
  covariate in predicating response. The response
  data are shown in next page with Y1 indicating
  relapse, death, or major intervention, such as
  bone marrow transplant before Day 90. Is there
  any evidence that administration of asODN is
  associated with a decreased relapse rate?

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Effect Selection Methods

• Statistical model selection will facilitate
  selection and screening of explanatory variables
  from a sets of candidate variables. The commonly
  used model selection method include
• Backward selection starting with all candidate
  variables and testing them one by one for
  statistical significance, deleting any that are
  not significant.
• Forward selection starting with no variables in
  the model, trying out the variables one by one
  and including them if they are 'statistically
  significant'.
• Stepwise selection A combination of both
  methods. Select a most significant variable from
  the candidate pool and remove this variable if
  its not significant in the joint model. And
  repeat this process step by step for all
  remaining variables.

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Flow Chart of Forward Selection
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Multicollinearity

• Multicollinearity occurs when two or more
  explanatory variables in a multiple regression
  model are highly correlated. In other words,
  there is redundant explanatory variables in the
  multiple regression models.
• Multicollinearity can cause problematic estimate
  in the individual effects. A high degree of
  multicollinearity can also cause computer
  software packages to be unable to perform the
  matrix inversion that is required for computing
  the regression coefficients, or it may make the
  results of that inversion inaccurate.
• Note that in statements of the assumptions
  underlying regression analyses such as ordinary
  least squares, the phrase "no multicollinearity"
  is sometimes used to mean the absence of perfect
  multicollinearity, which is an exact
  (non-stochastic) linear relation among the
  regressors.

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Detection of Multicollinearity

• Large changes in the estimated regression
  coefficients when a predictor variable is added
  or deleted
• Tests of the individual effects of affected
  variables are not significant, but a global test
  of overall model is significant (using an
  F-test).
• Use variance inflation factor (VIF) to detect
  multicollinearity Regress a explanatory variable
  on all the other explanatory variables. A high
  coefficient of determination, r2, indicates the
  regressed explanatory variable was highly
  corrected with other explanatory variables. A
  tolerance1-r2. VIF1/tolerance. A tolerance of
  less than 0.20 or 0.10 or a VIF of 5 or 10 and
  above indicates a multicollinearity problem.

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Caveats

• Sometimes logistic regression is carried out when
  a dependent variable is dichotomized. It is
  important that the cut point is not derived by
  direct examination of the data for example to
  find a gap in the data which maximizes the
  discrimination between the selected groups as
  this can lead to biased results. It is bests if
  there are a priori grounds for choosing a
  particular cut point.

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References

• Common Statistical Methods for Clinical Research
  2nd Edition by Glenn Walker
• Logistic Regression Using The SAS System by Paul
  Allison
• Medical Statistics by Campbell et al.