medical statistics as a science. Меdical statistics: to do this we must assume that all data is...

Medical Medical StatisticsStatistics

as a scienceas a science

МеМеdicaldical Statistics Statistics: :

To do this we must assume that all data is randomly sampled from an infinitely large population, then analyse this sample and use results to make inferences about the population

Extrapolate from data collected to make general conclusions about larger population from which data sample was derived

Allows general conclusions to be made from limited amounts of data

Statistical Analysis in a

Simple ExperimentHalf the subjects

receive one treatment and the other half

another treatment

(usually placebo)Define population of

interestUse statistical techniques to make inferences about the distribution of the variables in the general population and about the effect of the treatment

Measure baseline

variables in each group(e.g. age,

Apache II to ensure

randomisation successful)

Randomly select sample of subjects to study(exclusion criteria but define a precise patient population)

OutlineOutline

PowerPower Basic Sample Size InformationBasic Sample Size Information Examples (see text for more)Examples (see text for more) Changes to the basic formulaChanges to the basic formula Multiple comparisonsMultiple comparisons Poor proposal sample size Poor proposal sample size

statementsstatements Conclusion and ResourcesConclusion and Resources

Тypes of descriptive statistics: Тypes of descriptive statistics:

Measures of

central tendency

Graphs

Measures of variabi

lity

Categorical data: values belong to categories

DataNominal data: there is no natural order to the categoriese.g. blood groups

Numerical data: the value is a number(either measured or counted)

DataOrdinal data: there is natural order e.g. Adverse Events (Mild/Moderate/Severe/Life Threatening)

DataData

Categorical data:Categorical data: values belong to categories values belong to categories Nominal dataNominal data:: there is no natural order to the there is no natural order to the

categoriescategoriese.g. blood groupse.g. blood groups

Ordinal dataOrdinal data:: there is natural order e.g. Adverse there is natural order e.g. Adverse Events (Mild/Moderate/Severe/Life Threatening)Events (Mild/Moderate/Severe/Life Threatening)

Binary dataBinary data:: there are only two possible categories there are only two possible categoriese.g. alive/deade.g. alive/dead

Numerical data:Numerical data: the value is a number the value is a number(either measured or counted)(either measured or counted) Continuous dataContinuous data:: measurement is on a continuum measurement is on a continuum

e.g. height, age, haemoglobine.g. height, age, haemoglobin

Discrete dataDiscrete data:: a “count” of events e.g. number of a “count” of events e.g. number of pregnanciespregnancies

Descriptive StatisticsDescriptive Statistics::

concerned with summarising or concerned with summarising or describing a sample eg. mean, describing a sample eg. mean, medianmedian

Inferential StatisticsInferential Statistics::

concerned with generalising from a concerned with generalising from a sample, to make estimates and sample, to make estimates and inferences about a wider population inferences about a wider population eg. T-Test, Chi Square testeg. T-Test, Chi Square test

1)Basic requirement of

medical research

Why we need to study

statistics?

Why we need to study

statistics?

3)Data manage

ment and

treatment

2)Update your medical knowledge.

Statistical TermsStatistical Terms MeanMean:: the average of the data the average of the data

sensitive to outlying data sensitive to outlying data MedianMedian:: the middle of the data the middle of the data

not sensitive to outlying data not sensitive to outlying data ModeMode:: most commonly occurring value most commonly occurring value RangeRange:: the spread of the data the spread of the data IQ rangeIQ range:: the spread of the data the spread of the data

commonly used for skewed data commonly used for skewed data Standard deviationStandard deviation:: a single number which a single number which

measures how much measures how much the the observations vary around the meanobservations vary around the mean

Symmetrical dataSymmetrical data:: data that follows normal data that follows normal distribution distribution (mean=median=mode)(mean=median=mode)

report mean & standard report mean & standard deviation & deviation & nn

Skewed dataSkewed data:: not normally distributed not normally distributed (mean (meanmedian median mode) mode) report median & IQ Range report median & IQ Range

Standard Normal Standard Normal DistributionDistribution

Standard Normal Standard Normal DistributionDistribution

Mean +/- 1 SD encompasses 68% of observations

Mean +/- 2 SD encompasses 95% of observations

Mean +/- 3SD encompasses 99.7% of observations

15

1. Experimental Design1. Experimental Design Convenience SamplingConvenience Sampling

Use results that are easy to getUse results that are easy to get

16

1. Experimental Design1. Experimental Design Stratified SamplingStratified Sampling

Draw a sample from each stratumDraw a sample from each stratum

Basic conceptsBasic concepts

Homogeneity: All

individuals have

similar values or belong to

same category.

Example: all individuals are Chinese, women, middle age (30~40 years old), work in a textile mill ---- homogeneity in nationality, gender, age and occupation.

Variation:the differences in height, weight…

Steps in Statistical Steps in Statistical TestingTesting Null hypothesisNull hypothesis

Ho: there is no difference between the Ho: there is no difference between the groupsgroups

Alternative hypothesisAlternative hypothesisH1: there is a difference between the groupsH1: there is a difference between the groups

Collect dataCollect data

Perform test statistic eg T test, Chi squarePerform test statistic eg T test, Chi square

Interpret P value and confidence intervalsInterpret P value and confidence intervals

P value P value 0.05 Reject Ho 0.05 Reject Ho

P value > 0.05 Accept HoP value > 0.05 Accept Ho

Draw conclusionsDraw conclusions

Population and sample

Population: The whole collection

of individuals

that one intends to

study

Sample:A

representative part of

the populati

on.

Randomization: An

important way to make the sample

representative

ProbabilityProbability

Measure the possibility of Measure the possibility of occurrence of a random event.occurrence of a random event.

A : random eventA : random event P(A) : Probability of the random P(A) : Probability of the random

event Aevent A

P(A)=1 , if an event always occurs.P(A)=1 , if an event always occurs.

P(A)=0, if an event never occurs.P(A)=0, if an event never occurs.

21

2. 2. Descriptive Statistics & Descriptive Statistics & DistributionsDistributions

ParameterParameter: population quantity: population quantity StatisticStatistic: summary of the sample: summary of the sample Inference for parametersInference for parameters: use sample: use sample Central TendencyCentral Tendency

Mean (average)Mean (average) Median (middle value)Median (middle value)

VariabilityVariability Variance: measure of variationVariance: measure of variation Standard deviation (sd): square root of Standard deviation (sd): square root of

variancevariance Standard error (se): sd of the estimateStandard error (se): sd of the estimate Median, quartiles, min., max, range, boxplotMedian, quartiles, min., max, range, boxplot

ProportionProportion

22


Normal distributionNormal distribution

23


Standard normal distribution: Standard normal distribution: Mean 0, variance 1Mean 0, variance 1

24


Z-test for means Z-test for means T-test for means if sd is unknownT-test for means if sd is unknown

Meaning of PMeaning of P P Value: the probability of P Value: the probability of

observing a result as extreme or observing a result as extreme or more extreme than the one more extreme than the one actually observed from chance actually observed from chance alonealone

Lets us decide whether to reject or Lets us decide whether to reject or accept the null hypothesisaccept the null hypothesis

P > 0.05P > 0.05 Not significantNot significant P = 0.01 to 0.05P = 0.01 to 0.05 SignificantSignificant P = 0.001 to 0.01P = 0.001 to 0.01 Very significantVery significant P < 0.001P < 0.001 Extremely significantExtremely significant

26

3. 3. Inference for MeansInference for Means

Click ‘Statistics’ to select the

statistical procedure.

Click ‘File’ to open the SAS data set.

Click ‘File’ to import data and create

the SAS data set.

Click ‘Solution’ to create a

project to run statistical test

27


Mann-Whitney U-Test (Wilcoxon Rank-Mann-Whitney U-Test (Wilcoxon Rank-Sum Test)Sum Test)

Nonparametric alternative to two-sample t-Nonparametric alternative to two-sample t-testtest

The populations don’t need to be normalThe populations don’t need to be normal HH00: The two samples come from populations : The two samples come from populations

with equal medianswith equal medians HH11: The two samples come from populations: The two samples come from populations

with different medianswith different medians

28


Mann-Whitney U-Test ProcedureMann-Whitney U-Test Procedure Temporarily combine the two samples Temporarily combine the two samples

into one big sample, then replace each into one big sample, then replace each sample value with its rank sample value with its rank

Find the sum of the ranks for either Find the sum of the ranks for either one of the two samplesone of the two samples

Calculate the value of the Calculate the value of the z z test test statistic statistic

T TestT Test T test checks whether T test checks whether twotwo samples are likely to have come samples are likely to have come

from the same or different populationsfrom the same or different populations Used on continuous variablesUsed on continuous variables Example: Age of patients in the APC study (APC/placebo)Example: Age of patients in the APC study (APC/placebo)

PLACEBO: PLACEBO: APC: APC: mean age 60.6 yearsmean age 60.6 years mean age 60.5 yearsmean age 60.5 years

SD+/- 16.5SD+/- 16.5 SD +/- 17.2SD +/- 17.2 n= 840n= 840 n= 850n= 850 95% CI 59.5-61.795% CI 59.5-61.7 95% CI 59.3-61.795% CI 59.3-61.7

What is the P value?What is the P value? 0.010.01 0.050.05 0.100.10 0.900.90 0.990.99

P = 0.903 P = 0.903 not significant not significant patients from the same patients from the same populationpopulation(groups designed to be matched by randomisation so no (groups designed to be matched by randomisation so no surprise!!)surprise!!)

T Test: SAFE “Serum T Test: SAFE “Serum Albumin”Albumin”

Q: Are these albumin levels different?Q: Are these albumin levels different?Ho = Levels are the same (any difference is Ho = Levels are the same (any difference is there by chance)there by chance)H1 =Levels are too different to have occurred H1 =Levels are too different to have occurred purely by chancepurely by chance

Statistical test:Statistical test: T test T test P < 0.0001 (extremely P < 0.0001 (extremely significant)significant)Reject null hypothesis (Ho) and accept alternate Reject null hypothesis (Ho) and accept alternate hypothesis (H1) hypothesis (H1) ie. 1 in 10 000 chance that these samples are ie. 1 in 10 000 chance that these samples are both from the same overall group therefore we both from the same overall group therefore we can say they are very likely to be differentcan say they are very likely to be different

PLACEBOPLACEBO ALBUMIN ALBUMIN

nn 35003500 3500 3500

meanmean 2828 30 30

SDSD 1010 10 10

95% CI95% CI 27.7-28.327.7-28.3 29.7-30.3 29.7-30.3

Effect of Sample Size Effect of Sample Size ReductionReduction

smaller sample size (one tenth smaller)smaller sample size (one tenth smaller) causes wider CI (less confident where mean causes wider CI (less confident where mean

is)is) P = 0.008 (i.e. approx 0.01 P = 0.008 (i.e. approx 0.01 P is significant P is significant

but less so)but less so) This sample size influence on ability to find This sample size influence on ability to find

any particular difference as statistically any particular difference as statistically significant is a major consideration in study significant is a major consideration in study designdesign

PLACEBOPLACEBO ALBUMIN ALBUMIN

nn 350350 350 350

meanmean 2828 30 30

SDSD 1010 10 10

95% CI95% CI 27.0-29.027.0-29.0 29.0-31.0 29.0-31.0

Reducing Sample Size Reducing Sample Size (again)(again)

using even smaller sample size (now 1/100)using even smaller sample size (now 1/100) much wider confidence intervalsmuch wider confidence intervals p=0.41 (not significant anymore)p=0.41 (not significant anymore) SMALLER STUDY has LOWER POWER to SMALLER STUDY has LOWER POWER to

find any particular difference to be statistically find any particular difference to be statistically significant (mean and SD unchanged)significant (mean and SD unchanged)

POWER: POWER: the ability of a study to detect an the ability of a study to detect an actual effect or differenceactual effect or difference

PLACEBOPLACEBO ALBUMIN ALBUMINnn 3535 35 35

meanmean 2828 30 30

SDSD 1010 10 10

95% CI95% CI 24.6-31.424.6-31.4 26.6-33.4 26.6-33.4

33


Mann-Whitney U-Mann-Whitney U-Test, ExampleTest, Example

Numbers in Numbers in parentheses are their parentheses are their ranks beginning with ranks beginning with a rank of 1 assigned a rank of 1 assigned to the lowest value of to the lowest value of 17.7.17.7.

RR11 and and RR22: sum of : sum of ranksranks

34

3. 3. Inference for MeansInference for Means Hypothesis: The group means are differentHypothesis: The group means are different

HHoo: Men and women have same median BMI’s: Men and women have same median BMI’s HH11: Men and women have different median : Men and women have different median

BMI’sBMI’s

p-valuep-valuethus we do not reject Hthus we do not reject H00 at at =0.05.=0.05.

There is no significant difference in BMI There is no significant difference in BMI between men and women.between men and women.

1 1 2( 1) 13(13 12 1)169

2 2R

n n n

1 2 1 2( 1) (13)(12)(13 12 1)18.385

12 12R

n n n n

187 1690.98

18.385R

R

Rz

35


SAS Programming for Mann-Whitney U-SAS Programming for Mann-Whitney U-Test ProcedureTest Procedure

Data steps Data steps :: The same as slide 21.The same as slide 21. Procedure steps Procedure steps :: Click ‘Click ‘SolutionsSolutions’ Click ‘’ Click ‘AnalysisAnalysis’ Click ‘’ Click ‘AnalystAnalyst’ ’

Click ‘Click ‘FileFile’ Click ‘’ Click ‘Open By SAS NameOpen By SAS Name’ ’

Select the SAS data set and Click ‘Select the SAS data set and Click ‘OKOK’ ’

Click ‘Click ‘StatisticsStatistics’ Click ‘ ’ Click ‘ ANOVAANOVA’ ’

Click ‘Click ‘Nonparametric One-Way ANOVANonparametric One-Way ANOVA’ ’

Select the ‘Select the ‘DependentDependent’ and ‘’ and ‘IndependentIndependent’ variables respectively ’ variables respectively

and choose the interested test Click ‘and choose the interested test Click ‘OKOK’’

36




Click ‘File’ to open the SAS data

set.

Select the dependent and independent variables:

37


Notation for paired t-testNotation for paired t-test dd = individual difference between the two = individual difference between the two values of a single matched pair values of a single matched pair µµdd = = mean value of the differences mean value of the differences dd for the for the population of paired data population of paired data = = mean value of the differences mean value of the differences dd for the for the

paired sample data paired sample data ssdd = = standard deviation of the differences standard deviation of the differences dd

for the paired sample datafor the paired sample data nn = number of pairs = number of pairs

d

d

38

3. 3. Inference for MeansInference for Means Example: Systolic Blood PressureExample: Systolic Blood Pressure

OC:OC: Oral contraceptiveOral contraceptive

ID Without OC’s With OC’s Difference

1 115 128 13

2 112 115 3

3 107 106 -1

4 119 128 9

5 115 122 7

6 138 145 7

7 126 132 6

8 105 109 4

9 104 102 -2

10 115 117 2

39

3. 3. Inference for MeansInference for Means Hypothesis: The group means are Hypothesis: The group means are

differentdifferent HHoo: vs. H: vs. H11: : Significance level: Significance level: = 0.05 = 0.05 Degrees of freedom (df): Degrees of freedom (df): Test statisticTest statistic

P-value: 0.009, thus reject HP-value: 0.009, thus reject Hoo at at =0.05=0.05 The data support the claim that oral The data support the claim that oral

contraceptives affect the systolic bp.contraceptives affect the systolic bp.

0d 0d

32.310/57.4

8.4

/

ns

dt

d

d

91n

40

3. 3. Inference for MeansInference for Means Confidence interval for matched pairsConfidence interval for matched pairs

100(1-100(1-)% CI:)% CI:

95% CI for the mean difference of the systolic 95% CI for the mean difference of the systolic bp:bp:

(1.53, 8.07)(1.53, 8.07)

n

std

n

std d

nd

n 1,2/1,2/ ,

27.38.410

57.426.28.4

109,025.0 ds

td

41




Click ‘File’ to open the SAS data

set.

Put the two group variables into ‘Group 1’ and ‘Group 2’

Chi Square TestChi Square Test Proportions or frequenciesProportions or frequencies Binary data e.g. alive/deadBinary data e.g. alive/dead PROWESS Study: Primary endpoint: 28 day PROWESS Study: Primary endpoint: 28 day

all cause mortalityall cause mortalityALIVEALIVE DEAD TOTAL % DEAD DEAD TOTAL % DEAD

PLACEBO 581 (69.2%) 259 (30.8%)PLACEBO 581 (69.2%) 259 (30.8%) 840 (100%) 30.8840 (100%) 30.8

DEADDEAD 640 (75.3%) 640 (75.3%) 210 (24.7%) 210 (24.7%) 850 (100%) 24.7850 (100%) 24.7

TOTALTOTAL 1221 (72.2%) 1221 (72.2%) 469 (27.8%) 469 (27.8%) 1690 (100%)1690 (100%)

Perform Chi Square test Perform Chi Square test P = 0.006 (very significant) P = 0.006 (very significant) 6 in 1000 times this result could happen by chance6 in 1000 times this result could happen by chance 994 in 1000 times this difference was not by chance 994 in 1000 times this difference was not by chance variation variation

Reduction in death rate = 30.8%-24.7%= 6.1% Reduction in death rate = 30.8%-24.7%= 6.1% ie 6.1% less likely to die in APC group ie 6.1% less likely to die in APC group

Reducing Sample SizeReducing Sample Size Same results but using much smaller sample size (one tenth)Same results but using much smaller sample size (one tenth)

ALIVEALIVE DEAD TOTAL % DEAD DEAD TOTAL % DEAD

PLACEBO 58 (69.2%) 26 (30.8%) 84 (100%)PLACEBO 58 (69.2%) 26 (30.8%) 84 (100%) 30.8 30.8

DEADDEAD 64 (75.3%) 64 (75.3%) 21 (24.7%) 85 (100%) 21 (24.7%) 85 (100%) 24.7 24.7

TOTALTOTAL 122 (72.2%) 122 (72.2%) 47 (27.8%) 169 (100%) 47 (27.8%) 169 (100%)

Reduction in death rate = 6.1% (still the same)Reduction in death rate = 6.1% (still the same) Perform Chi Square test Perform Chi Square test P = 0.39 P = 0.39 39 in 100 times this difference in mortality could have 39 in 100 times this difference in mortality could have happened by chance therefore results not significant happened by chance therefore results not significant

Again, power of a study to find a difference depends a lot Again, power of a study to find a difference depends a lot on sample size for binary data as well as continuous data on sample size for binary data as well as continuous data

SummarySummary

Size matters=BIGGER IS BETTERSize matters=BIGGER IS BETTER Spread matters=SMALLER IS Spread matters=SMALLER IS

BETTERBETTER Bigger difference=EASIER TO FINDBigger difference=EASIER TO FIND Smaller difference=MORE Smaller difference=MORE

DIFFICULT TO FINDDIFFICULT TO FIND To find a small difference you need To find a small difference you need

a big studya big study

medical statistics as a science. Меdical statistics: to do this we must assume that all data is...

Documents

data sample

data sensitive

haemoglobindiscrete

data management

categoriesdatanominal

categoriesnominal data

meansymmetrical data

counteddataordinal data