Paper: Regression Analysis III

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Subject: Statistics

Paper: Regression Analysis III

Module: Prospective and Retrospective Studies

Regression Analysis III 1 / 18

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Development Team

Principal investigator: Dr. Bhaswati Ganguli,Professor, Department of Statistics, University of Calcutta

Paper co-ordinator: Dr. Bhaswati Ganguli,Professor, Department of Statistics, University of Calcutta

Content writer: Sayantee Jana, Graduate student, Department of Mathematics and Statistics, McMaster University Sujit Kumar Ray,Analytics professional, Kolkata

Content reviewer: Department of Statistics, University of Calcutta

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Study designs

Q: What are the different types of study design ? Q: What do they mean?

1. RetrospectiveStudy design

A study design that traces back into the past.

2. Prospectivestudy design

A study design that looks into the future.

Regression Analysis III Introduction to Study designs 3 / 18

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Study designs

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Study designs

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Study designs

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Study designs

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Study designs

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Types of Studies

Case-control study

A study which uses a retrospective design to trace back the past conditions of its sample units is called a case-control study.

I Case-control studies are very common in epidemiological or clinical studies.

I The sampled units under the diseased group are calledcases and the sampled units in the non-diseased group are called controls.

I When the two samples are matched in a case-control study it is called amatched case-control study.

I But when the two samples of cases and controls are not matched i.e. they are independently sampled, then it is called an unmatched case-control study.

Regression Analysis III Types of Studies 4 / 18

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Types of Studies

Case-control study

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Types of Studies

Case-control study

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Types of Studies

Case-control study

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Example of a retrospective study

Problem: To establish association between smoking and lung-cancer. ¹

Study design: In a data reported in the British Medical Journal in 1950, 709 cases of lung-cancer and 709 controls were sampled from 20 hospitals in London, England and were interviewed about their smoking behaviour.

The cases were sampled from patients who were admitted with lung cancer in the preceding year and the controls were sampled from non cancer patients at the same hospital of the same gender and within the same 5-year grouping of age.

A smoker was defined as a person who had smoked at least one cigarette a day for at least a year.

1Agresti, A. (2014). Categorical data analysis. John Wiley Sons.

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Example

This is a matched case-control study.

Y: occurrence of lung-cancer X: smoking behaviour

Table:Contingency table of smoking by lung cancer.

Lung Cancer Smoker Cases Controls

Yes 688 650

No 21 59

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Types of Studies continued ...

I A study design that follows its experimental subjects into the future to observe whether they develop the disease or not, is called a prospective study design.

I Prospective studies are of two types :

I Cohort studieswhere the subject decides whether they want to be exposed to a certain condition (of the predictor variable) or not.

I Cross-sectional studieswhere the samples are classified into the different exposure conditions during the design stage of the experiment.

I Cohort and cross-sectional studies are very common in clinical studies.

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Example of a prospective study design

Objective: The Physicians’ Health Study Research Group at Harvard Medical School conducted a study to find out whether regular aspirin intake reduces mortality from cardiovascular disease.²

Study design: It was a blinded 5-year randomized study where the physicians in the study took either one aspirin tablet or a placebo every alternate day. At then end of the study it as observed how many subjects had heart attack, fatal or non-fatal.

This is an example of a cross-sectional study.

2Agresti, A. (2014). Categorical data analysis. John Wiley

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Example

Y: Had heart attack during the course of the study X: Aspirin use

Table:Contingency table of Aspirin use and Myocardial Infarction .

Myocardial Infarction Smoker Heart attack No attack

Aspirin 104 10845

Placebo 189 10933

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Invariance of OR to study design

Let us consider a contingency table of probabilities of smoking vs.

lung cancer.

Also let us consider the example from the perspective of a prospective as well as a retrospective study design.

Y: occurrence of lung-cancer X: smoking behaviour Table:Contingency table of smoking by lung cancer.

Lung Cancer

Smoker Yes No

Yes π11 π12

No π21 π22

Regression Analysis III Invariance of OR to study design 10 / 18

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lung cancer.

Lung Cancer

Smoker Yes No

Yes π11 π12

No π21 π22

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lung cancer.

Lung Cancer

Smoker Yes No

Yes π11 π12

No π21 π22

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OR for Prospective study design

I Case I : Let us assume a prospective study design for the example just stated.

I So this means a population of smokers and non-smokers were followed into the future to find out who develops lung cancer (L.C.) and who does not, and then the corresponding

probabilities were tabulated.

I log of odds of developing L.C. among smokers = log^π_π¹¹

12

I log of odds of developing L.C among non-smokers = log^π_π²¹

22

I log (Odds Ratio) = log^π_π¹¹

12 - log^π_π²¹

22

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12

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12 - log^π_π²¹

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12

22

12 - log^π_π²¹

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12

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12 - log^π_π²¹

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12

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12 - log^π_π²¹

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OR for Retrospective study design

I Case II : Now let us assume a retrospective study design for the same example.

I So this means a population of lung cancer patients were traced back in time to find out who were smokers and who were not.

I log of odds that a lung cancer patient was a smoker = log^π_π¹¹

21

I log of odds that a non lung cancer patient was a smoker = log^π_π¹²

22

21 - log^π_π¹²

22

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21

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21 - log^π_π¹²

22

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21

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21 - log^π_π¹²

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21

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21 - log^π_π¹²

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21

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21 - log^π_π¹²

22

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I Hence it is quite obvious that both the designs lead us to the same population Odds ratio.

I Thus while calculating Odds Ratio it does not matter how the sampling can be drawn.

I We would reach the same conclusion for sample Odds ratio also if we consider a contingency table of sample counts.

I Hence Odds Ratio is invariant to study design.

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Summary

I There are two types of study designs :

I retrospective - a study design that looks into the past.

I prospective - a study design that looks into the future.

I Retrospective studies : Case-control studies

I Prospective studies : Cohort studies and Cross-sectional studies.

I Odds Ratio is invariant to study design.

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Summary

Regression Analysis III Summary 14 / 18

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Summary

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Summary

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Summary

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Summary

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Example 1 ³

## A cross sectional study investigating the relationship

## between dry cat food (DCF) and feline urologic syndrome

## (FUS) was conducted (Willeberg 1977). Counts of

## individuals in each group were as follows:

## DCF-exposed cats (cases, non-cases) 13, 2163

## Non DCF-exposed cats (cases, non-cases) 5, 3349

## Outcome variable (FUS) as columns:

dat <- matrix(c(13,2163,5,3349), nrow = 2, byrow = TRUE) rownames(dat) <- c("DF+", "DF-");

colnames(dat) <- c("FUS+", "FUS-"); dat

epi.2by2(dat = as.table(dat), method = "cross.sectional", conf.level = 0.95, units = 100, outcome = "as.columns")

3help file of the function epi.2by2 in the library epiR, R Documentation

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Example 1 contd ...

## Outcome variable (FUS) as rows:

dat <- matrix(c(13,5,2163,3349), nrow = 2, byrow = TRUE) rownames(dat) <- c("FUS+", "FUS-");

colnames(dat) <- c("DF+", "DF-"); dat

epi.2by2(dat = as.table(dat), method = "cross.sectional", conf.level = 0.95, units = 100, outcome = "as.rows")

## Odds Ratio remains the same even with change of rows

## and columns. This indicates odds ratio remains

## unchanged even when we interchange response and

## predictor.

Regression Analysis III Sample R script to complement this module 16 / 18

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Example 2 ⁴

## Prospective study babyfood[c(1,3),]

# the log-odds of having a respiratory disease for

# breastfed infants,are

# log47/447=-2.25

# the log-odds of having a respiratory disease for

# bottle-fed infants,are

# log47/447=-2.25

# log(OR)= -1.60- -2.25=0.65

# represents the increased risk of respiratory disease

# incurred by bottle feeding relative to breast feeding

4Faraway, J.J. (2006). Extending the Linear Model with R. Chapman Hall/CRC

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Example 2 contd ...

## Retrospective study

# Had this same data been from a retrospective study,

# the log of odds-ratio is the difference of log odds

# of feeding type given the disease status.

# log(OR)=log77/47-log381/447=log77/381-log47/447=0.65

# which is the same as the odds-ratio for the prospective

# study discussed above.

Regression Analysis III Sample R script to complement this module 18 / 18