Basic Statistics in Pharmaceutical Industry
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Basic Statistics in PharmaceuticalBasic Statistics in Pharmaceutical
IndustryIndustry
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ContentsContents
SampleSample
Bias and RandomizationBias and Randomization
Measure of Location/Central TendencyMeasure of Location/Central Tendency
Measure of DispersionMeasure of Dispersion
Hypothesis testingHypothesis testing
Type I and type II errorsType I and type II errors
PowerPower
PP--valuevalue
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Sources ofVariabilitySources ofVariability
which patients in the population get includedwhich patients in the population get included
in the studyin the study
which of the patients in the study get allocatedwhich of the patients in the study get allocated
to treatmentto treatment
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BiasBias
Bias is a measure of how much the sample is not aBias is a measure of how much the sample is not a
good representation of the population.good representation of the population.
Sources of bias:Sources of bias: selection biasselection bias
evaluator biasevaluator bias
measurement biasmeasurement bias
bias due to outcomebias due to outcome--related dropoutrelated dropout bias due to inconsistent study conductbias due to inconsistent study conduct
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RandomizationRandomization
Assures (with high probability) that treatment andAssures (with high probability) that treatment and
control groups are similar in all aspects except forcontrol groups are similar in all aspects except for
treatmenttreatment
Expect balance between known and unknown factorsExpect balance between known and unknown factors
Confidence that selection of patients into groups willConfidence that selection of patients into groups will
not be determined by a process that will give annot be determined by a process that will give an
advantage (bias) to one of the groupsadvantage (bias) to one of the groups Expect that group comparisons will be unbiasedExpect that group comparisons will be unbiased
Fundamental validity of statistical testsFundamental validity of statistical tests
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Measures ofMeasures of
Location/Central TendencyLocation/Central Tendency
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MeanMean
The mean of a sample of values is the
arithmetic average and is determined by
dividing the sum of the values by the numberof the values.
Mean = Sum of all the values / Total number of
Observations
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Example:50 Values of the Ritchie Index (MeasureExample:50 Values of the Ritchie Index (Measure
of Joint Stiffness) in 50 Untreated Patientsof Joint Stiffness) in 50 Untreated Patients
14 9 8 9 1 20 3 3 2 4
2 3 6 1 2 1116 2416 21
19 22 33 12 12 12 19 10 33 219 40 1 20 1 2 4 7 9 4
9 6 14 8 27 10 27 7 24 21
Here, Total number ofobservations are 50.
Mean = (14++21)/50 = 12.18
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MedianMedian
The median is the midpoint of the values when they arearranged in ascending order.
(If there are an even number of values there is no
midpoint value and the average of the two middle values
is taken).
If n is odd, then
Median = value of ((n+1)/2)th
term
If n is even, then
Median = mean of (n/2)th term and ((n/2)+1)th term
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1 1 1 1 2 2 2 2 2 3
3 3 4 4 4 6 6 7 7 8
8 9 9 9 9 10 10 11 12 1212 14 14 16 16 19 19 19 20 20
21 21 22 24 24 27 27 33 33 40
Ordered Ritchie Index ValuesOrdered Ritchie Index Values
Median = (9+10)/2 = 9.5
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ModeMode
The mode is the most repetitive or most
frequent value.
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1 1 1 1 2 2 2 2 2 3 3 34 4 4 6 6 7 7
8 8 9 9 9 9 10 10
11 12 12 12 14 1416 16 19 19 19 20 20
21 21 22 24 24
27 27 33 33 40
Here, modal value is 2
Ordered Ritchie Index ValuesOrdered Ritchie Index Values
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0
2
4
6
8
10
12
14
16
0 - 5 6 - 10 11 - 15 16 - 20 21 - 25 26 - 30 31 - 35 36 - 40
Values of the Ritchie Index
Arithmetic Mean - outlier prone
Median - only uses relative magnitudes
Mode - not necessarily central
Location = Central Tendency
freq
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Measures of Dispersion orMeasures of Dispersion orVariabilityVariability
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RangeRange
The range of a sample of values is the largestvalue minus the smallest value.
If the maximum and minimum value of data isIf the maximum and minimum value of data is
101 and 96 respectively then range is 101101 and 96 respectively then range is 101--96=596=5
Range is simple .. BUTRange is simple .. BUT
Only uses min and maxOnly uses min and max Gets larger as sample size increasesGets larger as sample size increases
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InterInter--quartile Rangequartile Range
The inter-quartile range of a sample of values is the
difference between the upper and lower quartiles.
Quartiles divide the ordered data into 4 parts. The
lower quartile is the value which is greater than ofthe sample and less than of the sample.
Conversely, the upper quartile is the value which is
greater than of the sample and less than of the
sample.
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1/4of50 = 12.5
3/4of50 = 37.5
1 1 1 1 2 2 2 2 2 3
3 3 4 4 4 6 6 7 7 88 9 9 9 9 10 10 11 12 12
12 14 14 16 16 19 19 19 20 20
2121 22 24 24 27 27 33 33 40
So, inter-quartile range = 37.5 12.5 = 25
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0
2
4
6
8
10
12
14
16
0 - 5 6 - 10 11 - 15 16 - 20 21 - 25 26 - 30 31 - 35 36 - 40
Values of the Ritchie Index
Lower quartile = 3.5
Upper quartile = 19
Inter-quartile range = 15.5
Freq
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Neither measure uses the numerical valuesNeither measure uses the numerical values -- only relativeonly relativemagnitudesmagnitudes
A measure which accounts for the values is theA measure which accounts for the values is the standardstandarddeviationdeviation
Consider the aspirin data from the new processConsider the aspirin data from the new process
96 97 100 101 101 (mean 99 mg)96 97 100 101 101 (mean 99 mg)
Determine deviations from meanDetermine deviations from mean --33 --2 1 2 22 1 2 2
Square , add, average and squareSquare , add, average and square--rootroot
standard deviation =standard deviation = 09219 !
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Confidence IntervalConfidence Interval
An interval summary that includes the concept of variation inAn interval summary that includes the concept of variation inpoint estimates.point estimates.
Confidence interval provides boundaries between which weConfidence interval provides boundaries between which weare relatively certain that the true value of a populationare relatively certain that the true value of a populationsummary lies.summary lies.
A common choice for the level of certainty is 95%. TA common choice for the level of certainty is 95%. Theheconstructedconstructed 95%95% CIs will cover the true populationCIs will cover the true populationmeasure of interest for 95% of all possible samples.measure of interest for 95% of all possible samples.
Since the realized sample is one but many possible samples,Since the realized sample is one but many possible samples,the constructed confidence interval based on a sample is onethe constructed confidence interval based on a sample is onebut many possible confidence intervals.but many possible confidence intervals.
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Making DecisionsMaking Decisions
Q: Given two samples from two populations, canQ: Given two samples from two populations, canwe tell, with certainty, if the two populations arewe tell, with certainty, if the two populations arethe same or different?the same or different?
A: We can not.A: We can not.
But, we can say whether they areBut, we can say whether they are likelylikely to be theto be thesame or different.same or different.
We are making decisions with calculated risks.We are making decisions with calculated risks.The calculated risks are specified beforehand inThe calculated risks are specified beforehand interms of probabilities.terms of probabilities.
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Philosophy ofStatistical TestingPhilosophy ofStatistical Testing
Assume that the opposite of what we wish toAssume that the opposite of what we wish toprove is true.prove is true.
Collect data.Collect data.
Under the assumption, compute how likely theUnder the assumption, compute how likely thedata are as we have observed.data are as we have observed.
If it is not very likely, conclude that the originalIf it is not very likely, conclude that the original
assumption is probably not true...assumption is probably not true... Otherwise, conclude that the data is consistentOtherwise, conclude that the data is consistent
with the assumption.with the assumption.
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Null and Alternative HypothesisNull and Alternative Hypothesis
Alternative hypothesis (HAlternative hypothesis (Haa) is what we would like to) is what we would like toconclude. For example, our treatment is better thanconclude. For example, our treatment is better thanour competitors; or our treatment is clinicallyour competitors; or our treatment is clinically
equivalent to our competitors.equivalent to our competitors. Null hypothesis (HNull hypothesis (H00) is usually the opposite of H) is usually the opposite of Haa..
Like prosecutors, we are trying to gather evidence toLike prosecutors, we are trying to gather evidence to
prove our cases. The defendant is assumed innocentprove our cases. The defendant is assumed innocentuntil proven guilty. Our job is to actively prove ouruntil proven guilty. Our job is to actively prove ourclaim of superiority or equivalence.claim of superiority or equivalence.
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Examples ofHExamples ofH00 and Hand Haa
Research hypothesis:Research hypothesis:
Zyvox increases % of individuals withZyvox increases % of individuals withmicrobiologic cures beyond that produced by themicrobiologic cures beyond that produced by the
comparator.comparator.Let pLet pZZ = % of individuals with microbiologic cures= % of individuals with microbiologic cures
when receiving Zyvox; pwhen receiving Zyvox; pCC = the corresponding= the corresponding
figure for the comparator.figure for the comparator.
So, we will testSo, we will testHH00: p: pZZ = p= pCCHHaa: p: pZZ p pCC or Hor Haa
**: p: pZZ > p> pCC
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ErrorsErrors
DecisionDecision
TruthTruth
HHaa
HH00
truetrue
Reject HReject H00
Correct decisionCorrect decision Type I error (Type I error (EE))
(false positive)(false positive)
Do Not Reject HDo Not Reject H00 Type II error (Type II error (FF))(false positive)(false positive)
Correct decisionCorrect decision
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Whose Risks?Whose Risks?
Type I errorType I error
False positiveFalse positive
Consumers riskConsumers risk
Regulators concernRegulators concern
Type II errorType II error
False negativeFalse negative
Sponsors riskSponsors risk Societal lossSocietal loss
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ConventionsConventions
In most situations in drug development, type I error is theIn most situations in drug development, type I error is themore grievous error.more grievous error.
Probability of a type I error is usually held at a specificProbability of a type I error is usually held at a specific
level (denoted by a), say 0.05. In other words, we arelevel (denoted by a), say 0.05. In other words, we areallowing a 5% chance to reject Hallowing a 5% chance to reject H00 even if Heven if H00 is true.is true.
Probability of a type II error is then a function of theProbability of a type II error is then a function of thesample size.sample size.
So, we want to choose an appropriate sample size toSo, we want to choose an appropriate sample size tocontrol the type II error rate at an acceptable level.control the type II error rate at an acceptable level.
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More ConventionsMore Conventions
Power is the complement of type II error rate. The lowerPower is the complement of type II error rate. The lowerthe type II error rate is, the higher the power, and the morethe type II error rate is, the higher the power, and the moresensitive the test is.sensitive the test is.
Power is the probability of getting a successful trial if thePower is the probability of getting a successful trial if thedrug does what we think it does.drug does what we think it does.
Among the threesome of type I error rate, power, andAmong the threesome of type I error rate, power, andsample size, we can calculate the third if we know thesample size, we can calculate the third if we know the
other two.other two. Many alternative hypotheses are composite, so we specifyMany alternative hypotheses are composite, so we specify
the desirable power to detect a clinically meaningfulthe desirable power to detect a clinically meaningfuldifference in the treatment effects.difference in the treatment effects.
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What Is PWhat Is P--Value?Value?
PP--values measure how consistent the data are with the nullvalues measure how consistent the data are with the nullhypothesis. They are basically the probabilities ofhypothesis. They are basically the probabilities ofobserving what we observed if the null hypothesis is true.observing what we observed if the null hypothesis is true.
Since samples can produce different summary results andSince samples can produce different summary results andPP--values are calculated from samples, Pvalues are calculated from samples, P--values can differvalues can differfrom sample to sample.from sample to sample.
If a PIf a P--value is less than the allowable type I error rate 5%,value is less than the allowable type I error rate 5%,
we will conclude that what we observed is not consistentwe will conclude that what we observed is not consistentwith the null hypothesis Hwith the null hypothesis H00. Therefore, the . Therefore, the proof byproof by
contradictioncontradiction approach leads to the rejection of H approach leads to the rejection of H00 andandthe acceptance of Hthe acceptance of Haa..
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Thank You!Thank You!
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Questions?Questions?