Statistics for clinical Statistics for clinical researchresearch

An introductory courseAn introductory course

Session 2Session 2

Comparing two groupsComparing two groups

Previous sessionPrevious session

Normal distributionNormal distribution

Standard Deviation (of measurements)Standard Deviation (of measurements)

Standard Error (of the mean)Standard Error (of the mean)

Confidence Interval of measurementsConfidence Interval of measurements

Confidence Interval of the meanConfidence Interval of the mean

Main overviewMain overview

Dealing with both Dealing with both MeansMeans and and ProportionsProportions

TwoTwo groups will be compared groups will be compared

Effect SizeEffect Size along with its along with its Confidence Confidence IntervalInterval (C.I.)(C.I.) will be calculated from data will be calculated from data

Remember theRemember the C.I. C.I. tells us about the tells us about the uncertaintyuncertainty of the of the effect sizeeffect size

The different calculations for effect sizesThe different calculations for effect sizes

MeansMeans

Means calculated from measured dataMeans calculated from measured data

Standard Deviation (of Measurements)Standard Deviation (of Measurements)

Standard Error (of the Mean)Standard Error (of the Mean)

Effect Size Effect Size == Difference in MeansDifference in Means

ProportionsProportions ProportionProportion

Binary outcome (e.g. yes/no)Binary outcome (e.g. yes/no) Number between 0 and 1Number between 0 and 1

2x2 table2x2 table

Effect sizesEffect sizes Risk Difference (RD); Relative Risk (RR); Risk Difference (RD); Relative Risk (RR);

Odds Ratio (OR)Odds Ratio (OR)

Group 1Group 1 Group 2Group 2

PositivePositive pp11 pp22

NegativeNegative nn11 nn22

Comparing two groupsComparing two groupsTwo proportionsTwo proportions

Risk DifferenceRisk Difference Number Needed to TreatNumber Needed to Treat Relative RiskRelative Risk Odds RatioOdds Ratio Fisher’s Exact ProbabilityFisher’s Exact Probability

Two meansTwo means The The tt-distribution-distribution Difference between meansDifference between means

Risk DifferenceRisk Difference Risk is a proportion (number between 0 Risk is a proportion (number between 0

and 1)and 1) Each group incorporate its own riskEach group incorporate its own risk Group 1: 15 people are given money…Group 1: 15 people are given money…

Happy Happy = 12= 12Not happy Not happy = 3= 3Total Total = 15= 15Risk of happiness = 12/15 = 0.8Risk of happiness = 12/15 = 0.8

Group 2: 10 people are not given money…Group 2: 10 people are not given money…Happy Happy = 5= 5Not happy Not happy = 5= 5Total Total = 10= 10Risk of happiness = 5/10 = 0.5Risk of happiness = 5/10 = 0.5

Risk DifferenceRisk Difference

Risk Difference (RD) is the risk of Risk Difference (RD) is the risk of one group subtracted from the risk one group subtracted from the risk of the other groupof the other group

RD = 0.8 – 0.5 = 0.3RD = 0.8 – 0.5 = 0.3

Excel file “TwoGroups.xls”Excel file “TwoGroups.xls”

Comparing two groupsComparing two groups

Two proportionsTwo proportions Risk DifferenceRisk Difference Number Needed to TreatNumber Needed to Treat Relative RiskRelative Risk Odds RatioOdds Ratio Fisher’s Exact ProbabilityFisher’s Exact Probability

Two meansTwo means The The tt-distribution-distribution Difference between meansDifference between means

Number Needed to TreatNumber Needed to Treat

NNT = 1 / Risk DifferenceNNT = 1 / Risk Difference

If RD = 0.21 (21%), then need to treat If RD = 0.21 (21%), then need to treat 100 to prevent 21 adverse events100 to prevent 21 adverse events

NNT = 1 / 0.21 = 5 (rounded up)NNT = 1 / 0.21 = 5 (rounded up)

5 need to be treated to prevent 1 5 need to be treated to prevent 1 additional adverse eventadditional adverse event

Excel file “TwoGroups.xls” Excel file “TwoGroups.xls”

Comparing two groupsComparing two groups

Two proportionsTwo proportions Risk DifferenceRisk Difference Number Needed to TreatNumber Needed to Treat Relative RiskRelative Risk Odds RatioOdds Ratio Fisher’s Exact ProbabilityFisher’s Exact Probability

Two meansTwo means The The tt-distribution-distribution Difference between meansDifference between means

Relative Risk (RR)Relative Risk (RR) Risk is a proportionRisk is a proportion

Each of the two groups has its own riskEach of the two groups has its own risk

Relative Risk (RR) is the ratio of two risksRelative Risk (RR) is the ratio of two risks

RR is mostly used for cohort studiesRR is mostly used for cohort studies

Ratios do not have a Normal distributionRatios do not have a Normal distribution

log(RR) has a Normal distributionlog(RR) has a Normal distribution

Confidence interval calculations require a Confidence interval calculations require a Normal distributionNormal distribution

Excel file “TwoGroups.xls”Excel file “TwoGroups.xls”

Relative Risk (RR)Relative Risk (RR)

If Confidence Interval…If Confidence Interval…

Contains 1:Contains 1: No difference in No difference in outcome between two groupsoutcome between two groups

<1:<1: Less risk in group 1 Less risk in group 1

>1:>1: Greater risk in group 1 Greater risk in group 1

Comparing two groupsComparing two groups

Two proportionsTwo proportions Risk DifferenceRisk Difference Number Needed to TreatNumber Needed to Treat Relative RiskRelative Risk Odds RatioOdds Ratio Fisher’s Exact ProbabilityFisher’s Exact Probability

Two meansTwo means The The tt-distribution-distribution Difference between meansDifference between means

Odds Ratio (OR)Odds Ratio (OR) Odds – the number who have an event Odds – the number who have an event

divided by the number who do notdivided by the number who do not Odds of an event occurring is obtained for Odds of an event occurring is obtained for

both groupsboth groups OR mostly used for case-control studiesOR mostly used for case-control studies Ratios are not Normally distributedRatios are not Normally distributed log(OR) has a Normal distributionlog(OR) has a Normal distribution Confidence Interval calculations require a Confidence Interval calculations require a

Normal distributionNormal distribution Extra: Logistic regression is typically used to Extra: Logistic regression is typically used to

adjust odds ratios to control for potential adjust odds ratios to control for potential confounding by other variablesconfounding by other variables

Excel file “TwoGroups.xls”Excel file “TwoGroups.xls”

Odds Ratio (OR)Odds Ratio (OR)

If Confidence Interval…If Confidence Interval…

Contains 1:Contains 1: No difference in No difference in outcome between two groupsoutcome between two groups

<1:<1: Odds in group 1 significantly Odds in group 1 significantly lessless

>1:>1: Odds in group 1 significantly Odds in group 1 significantly greatergreater

Comparing two groupsComparing two groups

Two proportionsTwo proportions Risk DifferenceRisk Difference Number Needed to TreatNumber Needed to Treat Relative RiskRelative Risk Odds RatioOdds Ratio Fisher’s Exact ProbabilityFisher’s Exact Probability

Two meansTwo means The The tt-distribution-distribution Difference between meansDifference between means

Fisher’s Exact TestFisher’s Exact Test

Determines if significant associations Determines if significant associations exist between group and outcomeexist between group and outcome

Used when sample sizes are smallUsed when sample sizes are small i.e. cell count < 5 in a 2x2 tablei.e. cell count < 5 in a 2x2 table

Alternative to the Chi-Square testAlternative to the Chi-Square test

Test only provides a p-value (no C.I.)Test only provides a p-value (no C.I.)

Probability of observing a result more Probability of observing a result more extreme than that observedextreme than that observed

Comparing two groupsComparing two groups

Two proportionsTwo proportions Risk DifferenceRisk Difference Number Needed to TreatNumber Needed to Treat Relative RiskRelative Risk Odds RatioOdds Ratio Fisher’s Exact ProbabilityFisher’s Exact Probability

Two meansTwo means The The tt-distribution-distribution Difference between meansDifference between means

The The tt-distribution-distribution

Population SD is unknown and Population SD is unknown and is estimated from the datais estimated from the data

Blue curve = Normal Blue curve = Normal distributiondistribution

Green = Green = tt-distribution with 1 -distribution with 1 degree of freedom (df)degree of freedom (df)

Red = Red = tt-distribution, 2 df-distribution, 2 df

Underlying theory of the Underlying theory of the tt-test-test

Comparing two groupsComparing two groups

Two proportionsTwo proportions Risk DifferenceRisk Difference Number Needed to TreatNumber Needed to Treat Relative RiskRelative Risk Odds RatioOdds Ratio Fisher’s Exact ProbabilityFisher’s Exact Probability

Two meansTwo means The The tt-distribution-distribution Difference between meansDifference between means

Difference between meansDifference between means

Two sampleTwo sample t t-test is used to test the -test is used to test the difference between two meansdifference between two means

Measurements must be considered Measurements must be considered Normally distributedNormally distributed

Quite powerful. A decision can be Quite powerful. A decision can be made with a small sample size…much made with a small sample size…much smaller than when compared to smaller than when compared to proportionsproportions

Excel file “TwoGroups.xls”Excel file “TwoGroups.xls”

Forest PlotForest Plot

Plot effect sizes with confidence intervalsPlot effect sizes with confidence intervals

Useful in comparing multiple effect sizesUseful in comparing multiple effect sizes

Go to applet on website:Go to applet on website:

http://www.materrsc.org/Course/CI_Diff.htmlhttp://www.materrsc.org/Course/CI_Diff.html

Additional topicsAdditional topics

Normality tests (e.g. Shapiro-Wilk)Normality tests (e.g. Shapiro-Wilk)

Test for equality of variances (e.g. Test for equality of variances (e.g. Bartlett’s test)Bartlett’s test)

tt-test for unequal variances-test for unequal variances

Paired t-test for dependent samplesPaired t-test for dependent samples

Comparing more than two groups Comparing more than two groups (e.g. one-way ANOVA)(e.g. one-way ANOVA)

Nonparametric testsNonparametric tests