Introduction to Univariate Statistics MSc in Pharmaceutical Medicine Statistics & Data Management...

Introduction to Univariate Statistics MSc in Pharmaceutical Medicine Statistics & Data Management Module Irene Rebollo-Mesa Senior Lecturer in Trials Kings Clinical Trials Unit, Biostatistics Institute of Psychiatry, Kings College London [email protected]

Outline 1Comparing 2 Independent Samples 2Comparing differences in a Paired Sample 3Compare Several Independent Samples 4Conclusions 2

1 Comparing 2 Independent Samples 3

Design or Aim of StudyType of Outcome Data/Assumptions Statistical Method COMPARE TWO INDEPENDENT SAMPLES Compare Two MeansContinuous, Normal dist.Independent Samples T-test Compare Two ProportionsCategorical, Binary, all >5Chi-squared test Compare Two ProportionsCategorical, Binary, some

Cocco G, Pandolfi S, Rousson V: Sufficient Weight Reduction Decreases Cardiovascular Complications in Diabetic Patients with the Metabolic Syndrome. A Randomized Study of Orlistat as an Adjunct to Lifestyle Changes (Diet and Exercise) Heart Drug 2005;5:68-74 90 patients with metabolic syndrome and diabetes Aged: >35 years; BMI: 31-40; LVEF: 42-50% Treatment: Daily exercise, diet and either Orlistat or Placebo Primary Outcome: Weight loss after diet of 6 months (kgs) Orlistat0.83.07.47.98.63.18.610.86.03.66.06.17.96.03.0 8.76.23.24.27.616.03.16.95.68.37.74.44.67.311.6 13.77.37.86.7-3.91.83.32.31.15.0-0.87.04.3-2.8-3.4 Placebo-0.91.53.45.65.26.42.95.63.71.72.23.85.50.74.6 1.42.0-0.24.91.95.72.04.53.44.53.82.92.53.31.5 5.94.94.5-2.1-0.52.21.6-0.60.0-1.56.31.7-6.2-1.9 Independent Samples t-test: Example 7

Independent Samples t-test: Example Histograms 8

Independent Samples t-test: Example Boxplots 9

Patients can be expected to respond differently Orlistat does not reduce weight in every patient Some patients who receive Placebo lose weight. The sample means for this experiment are : Orlistat: 5.41 kgs Placebo: 2.48 kgs NULL Hypothesis In the population of patients with metabolic syndrome and diabetes aged more than 35 the difference in mean weight changes at 8 months is zero kgs Patients can be expected to respond differently Does the difference in sample means, 2.93 kgs, provide evidence that the null hypothesis is not true? Independent Samples t-test: Example Rationale 10

The Two-sample t-test is based on the difference in sample means divided by the standard error (s.e.) of the difference in sample means: Weight Change Example: Difference in Sample Means: 5.41 -2.48= 2.93 kgs s.e (Difference in Sample Means) : 0.72 kgs t = 2.93/0.72 = 4.09 : p-value=0.000096 Independent Samples t-test: Example Calculation 11

Each sample is representative of the population There are no difference between the samples other than the treatments Measurements in each sample are independent of each other and of the measurements in the other sample Measurements are Normally distributed in the population The variances in each sample are the same If the assumptions of the test are not met, the p-value may be misleading 12 Independent Samples t-test: Assumptions

Design or Aim of StudyType of Outcome Data/Assumptions Statistical Method COMPARE TWO INDEPENDENT SAMPLES Compare Two MeansContinuous, Normal dist.Independent Samples T-test Compare Two Proportions Categorical, Binary, all >5Chi-squared test Compare Two ProportionsCategorical, Binary, some

Relative Risk between 2 groups: living vs. deceased. Most appropriate for prospective-cohort studies with complete follow up. H 0 : RR=1. RR=2.21 -> Kidney failure is 2.21 times higher in deceased donors than in living donors. Most appropriate for retrospective or case-control studies. Better mathematical properties, can be adjusted using log.reg. It approximates RR for small prev. H 0 : OR=1. OR=2.63 -> Among deceased donors the odds of kidney failure is 2.63 times higher than in living donors 17 Binary Outcome Data: Size of Effect Odds Ratio between 2 groups: living vs. deceased.

Design or Aim of StudyType of Outcome Data/Assumptions Statistical Method COMPARE TWO INDEPENDENT SAMPLES Compare Two MeansContinuous, Normal dist.Independent Samples T-test Compare Two ProportionsCategorical, Binary, all >5Chi-squared test Compare Two Proportions Categorical, Binary, some 5Chi-squared test Compare Two ProportionsCategorical, Binary, some

o Outcome V.: Urinary Excretion-->Continuous o Independent V.: 1 factor with two levels, within sbjs: MDI vs. DISK paired-samples t-test uses the following test statistic: t = sample mean (difference) / s.e. of sample mean (difference) o Under the Null Hypothesis (H 0 ) we expect to see a mean difference in urinary excretion close to zero in the sample. o The larger the sample mean difference (either positive or negative) the more our data appear to depart from the H 0. o The standard error (=standard deviation / n) of the sample mean tells us how far from zero we might expect the sample mean to be under the N Paired samples T-test: Example 31

Null Hypothesis: In the population of volunteers studied urinary excretion of Albuterol is the same whether delivered with MDI or DISK Does our sample mean of 1.18 provide any evidence that this is not true? Sample mean = -1.181; S.E. of sample mean = 0.459 Statistic:t = -1.181/0.459 = -2.574 Using a computer, or published tables, we obtain p=0.0329 (two-sided) i.e. if the H 0 were true there would only be a 3.3 % chance that we would see such a large, or larger, mean change in our sample. Paired samples T-test: Example 32

What Have We Learned? Every subject can be expected to respond differently. DISK does not increase the urinary excretion of every patient. The mean difference in urinary excretion was - 1.18 mmol/litre (significantly less on MDI) We can not be certain how new volunteers will respond to the same treatment. 33

Assumptions of paired-samples t-test The sample is representative of the population. Measurements in the sample are all independent of one another. Measurements are Normally distributed in the population. If the assumptions of a test are not met the p-value may not be correct. 34

Design or Aim of StudyType of Outcome Data/Assumptions Statistical Method COMPARE DIFFERENCES IN A PAIRED SAMPLE Test Mean DifferenceContinuous, Normal dist. for differences Paired (matched) Samples T-test Compare Two paired Proportions Categorical, BinaryMcNemars test Distribution of DifferencesOrdinal, symmetrical distributions Wilcoxon matched paired test Univariate Statistical Methods: Compare Two Paired Proportions 35

Binary outcome, Paired samples: McNemars Test Bronchodialator treatment and deaths from asthma: case-control study. BMJ 2005 Each patient who died was matched to a surviving patient, and risk factors looked at use of 2 Antagonist. H 0 :The prevalence of use of 2 Antagonist is the same among patients who died and patients who survived McNemars is based on discordant pairs. Assuming there is no association there should be as many of each (yes/no) & (no/yes) = 36

Binary outcome, Paired samples: McNemars Test Expected Frequency of Discordant pairs = (69+45)/2 = 57 = 37

Design or Aim of StudyType of Outcome Data/Assumptions Statistical Method COMPARE DIFFERENCES IN A PAIRED SAMPLE Test Mean DifferenceContinuous, Normal dist. for differences Paired (matched) Samples T-test Compare Two paired Proportions Categorical, BinaryMcNemars test Distribution of Differences Ordinal, symmetrical distributions Wilcoxon matched paired test Univariate Statistical Methods: Test Distribution of Differences in Paired Samples 38

39 CROSS-OVER TRIAL: Measure of Treatment Effect on proportion of days with headache H 0 =The distribution of differences is symmetrical about zero test based on probability ditribution of ranks of the differences If the null hypothesis is true a plus is as likely as a minus Probability(A>P)= Probability(P>A)=0.5 PlaceboActiveDifferenceSign A-P Rank 0.680.61 -0.07 - 10.5 0.961.00 0.04 + 6 0.850.72 -0.13 - 20 0.930.81 -0.12 - 17.5 0.350.26 -0.09 - 15 0.770.7 -0.07 - 10.5 0.740.61 -0.13 - 20 0.981.00 0.02 + 2 1.000.98 -0.02 - 2 0.280.16 -0.12 - 17.5 0.920.79 -0.13 - 20 1.000.98 -0.02 - 2 0.931.00 0.07 + 10.5 1.000.96 -0.04 - 6 0.450.49 0.04 + 6 0.820.74 -0.08 - 13.5 0.360.26 -0.1 - 16 1.000.97 -0.03 - 4 0.490.43 -0.06 - 8 0.740.66 -0.08 - 13.5 0.680.61 -0.07 - 10.5 Sample Mean-0.056 Ordinal Outcome Data: Wilcoxon Matched Pairs Test

-We rank the differences ignoring the sign -We sum the ranks of the positive and negative differences separately (excluding 0): T + & T - If the NH were true, we would expect the two rank sums to be about the same The test statistic, T, is the lesser of the sums. T + =206.5, T - =24.5. The smaller the T, the lower the probability of the data arising by chance. 40 Ordinal Outcome Data: Wilcoxon Matched Pairs Test

3 Compare Several Independent Samples 41

Design or Aim of StudyType of Outcome Data/Assumptions Statistical Method COMPARE SEVERAL INDEPENDENT SAMPLES Compare Several meansContinuous, Normal dist., Same Variance One-way Analysis of Variance (ANOVA) Compare Time to an Event in Several Groups SurvivalLogrank Test Univariate Statistical Methods: Compare Several Independent Samples 42

Design or Aim of StudyType of Outcome Data/Assumptions Statistical Method COMPARE SEVERAL INDEPENDENT SAMPLES Compare Several meansContinuous, Normal dist., Same Variance One-way Analysis of Variance (ANOVA) Compare Time to an Event in Several Groups SurvivalLogrank Test Univariate Statistical Methods: Compare Several Means 43

Compare Several Means: ANOVA An extension of t-test to compare 3 or more independent groups. H 0 : The samples for each group com from populations with same mean values. It provides one p value comparing all groups. Only if that is significant further contrasts are justified. ANOVA is based on partitioning variability : Between Group Variance: Variability (differences) between the groups Residual Variance: Remaining variability due to within group differences. 44

Statistic: The F ratio of the two variances if the groups are truly different Between-Group variability should be greater than the residual (Within-Group). Visual ANOVA" from the Wolfram Demonstrations Project http://demonstrations.wolfram.com/VisualANOVA/ ANOVA: The F ratio 45

ANOVA: The F ratio Hi BW Low Within Not all groups differ! 46

ANOVA: The F ratio Hi BW Hi Within 47

ANOVA: Example Use of RAB 753 in psoriasis a multi centre Phase 2 clinical trial Protocol: A double blind randomised placebo controlled parallel group study of 2 concentrations of topical RSB 753 (.5 and 1%) in patients with moderate to severe psoriasis. Primary Outcome Measure: PASI score (4 weeks score on table). Treatment Groups: Placebo,.5% and 1%. N=24 per group Placebo0.50%1% 4.243.2 43.93.2 4.33.93.6 4.53.34.1 4.23.93.8 4.83.53.1 4.842.9 44.44.8 3.64.9 4.64.43.4 4.743.5 3.83.72.8 4.73.52.7 4.64.74.8 4.643.2 4.33.53.2 4.643.7 43.64.5 4.34.43 4.742.8 4.84 4.93.53 43.62.8 434.8 48

ANOVA: Example 49

ANOVA: Example Group Means are different overall.

5 Conclusions The Design and Type of Data determine the proper statistical test to be used. Statistical analysis must be guided by hypothesis. Its not a fishing expedition Beware of Type I error. Specify clearly for your statistician: 1.Design 2.Hypothesis 3.Variables: type and scale of measurement 51

Introduction to Univariate Statistics MSc in Pharmaceutical Medicine Statistics & Data Management...

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