Medical Medical StatisticsStatistics

as a scienceas a science

Why Do Statistics?Why Do Statistics? Extrapolate from data collected to make Extrapolate from data collected to make

general conclusions about larger general conclusions about larger population from which data sample was population from which data sample was derivedderived

Allows general conclusions to be made Allows general conclusions to be made from limited amounts of datafrom limited amounts of data

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

Statistical AnalysisStatistical Analysisin a Simple Experimentin a Simple Experiment

Define population of interestDefine population of interest

Randomly select sample of subjects to studyRandomly select sample of subjects to study(clinical trials do not enrol a randomly selected sample of (clinical trials do not enrol a randomly selected sample of patients due to inclusion/exclusion criteria but define a patients due to inclusion/exclusion criteria but define a precise patient population)precise patient population)

Half the subjects receive one treatment and the other half Half the subjects receive one treatment and the other half another treatment (usually placebo)another treatment (usually placebo)

Measure baseline variables in each groupMeasure baseline variables in each group(e.g. age, Apache II to ensure randomisation successful)(e.g. age, Apache II to ensure randomisation successful)

Measure trial outcome variables in each group (e.g. mortality)Measure trial outcome variables in each group (e.g. mortality)

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

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

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

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

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

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 is)causes wider CI (less confident where mean 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

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