Transcript of Introduction to Univariate Statistics MSc in Pharmaceutical Medicine Statistics & Data Management...
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- 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
Irene.rebollo_mesa@kcl.ac.uk
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- Outline 1Comparing 2 Independent Samples 2Comparing differences
in a Paired Sample 3Compare Several Independent Samples
4Conclusions 2
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- 1 Comparing 2 Independent Samples 3
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- 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
- 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
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- Independent Samples t-test: Example Histograms 8
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- Independent Samples t-test: Example Boxplots 9
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- 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
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- 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
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- 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
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- 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.
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- 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
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- 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
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- 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
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- 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
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- 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
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- 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
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- Binary outcome, Paired samples: McNemars Test Expected
Frequency of Discordant pairs = (69+45)/2 = 57 = 37
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- 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
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- 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
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- -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
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- 3 Compare Several Independent Samples 41
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- 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
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- 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
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- 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
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- 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
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- ANOVA: The F ratio Hi BW Low Within Not all groups differ!
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- ANOVA: The F ratio Hi BW Hi Within 47
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- 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
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- ANOVA: Example 49
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- ANOVA: Example Group Means are different overall.
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- 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