Statistics for clinical
research
An introductory course
Session 2
Comparing two groups
Previous session

Normal distribution

Standard Deviation (of measurements)

Standard Error (of the mean)

Confidence Interval of measurements

Confidence Interval of the mean
Main overview

Dealing with both Means and Proportions

Two groups will be compared

Effect Size along with its Confidence
Interval (C.I.) will be calculated from data

Remember the C.I. tells us about the
uncertainty of the effect size

The different calculations for effect sizes
Means

Means calculated from measured data

Standard Deviation (of Measurements)

Standard Error (of the Mean)

Effect Size = Difference in Means
Proportions

Proportion




Binary outcome (e.g. yes/no)
Number between 0 and 1
2x2 table
Group 1
Group 2
Positive
p1
p2
Negative
n1
n2
Effect sizes

Risk Difference (RD); Relative Risk (RR);
Odds Ratio (OR)
Comparing two groups
Two proportions

Risk Difference

Number Needed to Treat

Relative Risk

Odds Ratio

Fisher’s Exact Probability
Two means

The t-distribution

Difference between means
Risk Difference

Risk is a proportion (number between 0
and 1)
Each group incorporate its own risk

Group 1: 15 people are given money…

Happy
= 12
Not happy = 3
Total
= 15
Risk of happiness = 12/15 = 0.8

Group 2: 10 people are not given money…
Happy
=5
Not happy = 5
Total
= 10
Risk of happiness = 5/10 = 0.5
Risk Difference

Risk Difference (RD) is the risk of
one group subtracted from the risk
of the other group

RD = 0.8 – 0.5 = 0.3

Excel file “TwoGroups.xls”
Comparing two groups
Two proportions





Risk Difference
Number Needed to Treat
Relative Risk
Odds Ratio
Fisher’s Exact Probability
Two means


The t-distribution
Difference between means
Number Needed to Treat

NNT = 1 / Risk Difference

If RD = 0.21 (21%), then need to treat
100 to prevent 21 adverse events

NNT = 1 / 0.21 = 5 (rounded up)

5 need to be treated to prevent 1
additional adverse event

Excel file “TwoGroups.xls”
Comparing two groups
Two proportions





Risk Difference
Number Needed to Treat
Relative Risk
Odds Ratio
Fisher’s Exact Probability
Two means


The t-distribution
Difference between means
Relative Risk (RR)

Risk is a proportion

Each of the two groups has its own risk

Relative Risk (RR) is the ratio of two risks

RR is mostly used for cohort studies

Ratios do not have a Normal distribution

log(RR) has a Normal distribution

Confidence interval calculations require a
Normal distribution

Excel file “TwoGroups.xls”
Relative Risk (RR)

If Confidence Interval…

Contains 1: No difference in
outcome between two groups

<1: Less risk in group 1

>1: Greater risk in group 1
Comparing two groups
Two proportions





Risk Difference
Number Needed to Treat
Relative Risk
Odds Ratio
Fisher’s Exact Probability
Two means


The t-distribution
Difference between means
Odds Ratio (OR)

Odds – the number who have an event
divided by the number who do not

Odds of an event occurring is obtained for
both groups

OR mostly used for case-control studies

Ratios are not Normally distributed

log(OR) has a Normal distribution

Confidence Interval calculations require a
Normal distribution

Extra: Logistic regression is typically used to
adjust odds ratios to control for potential
confounding by other variables

Excel file “TwoGroups.xls”
Odds Ratio (OR)

If Confidence Interval…

Contains 1: No difference in
outcome between two groups

<1: Odds in group 1 significantly
less

>1: Odds in group 1 significantly
greater
Comparing two groups
Two proportions





Risk Difference
Number Needed to Treat
Relative Risk
Odds Ratio
Fisher’s Exact Probability
Two means


The t-distribution
Difference between means
Fisher’s Exact Test

Determines if significant associations
exist between group and outcome

Used when sample sizes are small

i.e. cell count < 5 in a 2x2 table

Alternative to the Chi-Square test

Test only provides a p-value (no C.I.)

Probability of observing a result more
extreme than that observed
Comparing two groups
Two proportions





Risk Difference
Number Needed to Treat
Relative Risk
Odds Ratio
Fisher’s Exact Probability
Two means


The t-distribution
Difference between means
The t-distribution

Population SD is unknown and
is estimated from the data

Blue curve = Normal
distribution

Green = t-distribution with 1
degree of freedom (df)

Red = t-distribution, 2 df

Underlying theory of the t-test
Comparing two groups
Two proportions





Risk Difference
Number Needed to Treat
Relative Risk
Odds Ratio
Fisher’s Exact Probability
Two means


The t-distribution
Difference between means
Difference between means

Two sample t-test is used to test the
difference between two means

Measurements must be considered
Normally distributed

Quite powerful. A decision can be
made with a small sample size…much
smaller than when compared to
proportions

Excel file “TwoGroups.xls”
Forest Plot

Plot effect sizes with confidence intervals

Useful in comparing multiple effect sizes

Go to applet on website:
http://www.materrsc.org/Course/CI_Diff.html
Additional topics

Normality tests (e.g. Shapiro-Wilk)

Test for equality of variances (e.g.
Bartlett’s test)

t-test for unequal variances

Paired t-test for dependent samples

Comparing more than two groups
(e.g. one-way ANOVA)

Nonparametric tests