Course Notes - Department of Mathematics and Statistics

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Definitions IIPositive Predictive Value = Pr(A|B) The proportion of patientswith positive test results who are correctly diagnosed.Negative Predictive Value = Pr(Ā|¯B) The proportion of patientswith negative test results who are correctly diagnosed.Screening Programmes• A patient with certain symptoms consuled her doctor to be checkedfor a cancer, and she undergoes a biopsy.• With this test there is a probability of 0.90 that a woman withthe cancer shows a positive biopsy, and a probability of only 0.001that a healthy woman incorrectly shows a positive biopsy.• Historical information also suggests that the prevalence of thiscancer in the population is 1 in 10000.• Find the probability that a woman has the cancer given the biopsysays she does (i.e. does the biopsy diagnose true patient status?).• Let A be the event ‘woman has the cancer’ and B be the event‘biopsy is positive’.• Pr(A) = 1/10000 = 0.0001 (disease prevalence)• Pr(B| A) = 0.90 (conditional probability)• Pr(B| Ā)= 0.001 44
0.90BBiopsy +ve (true positive)0.00009A0.00010.10¯BBiopsy -ve (false negative)0.000010.99990.001BBiopsy +ve (false positive)0.00100Ā0.999¯BBiopsy -ve (true negative)0.99890Probability that the test is +ve = 0.00009 + 0.00100 = 0.00109Positive Predictive value• Find the positive predictive value Pr(A|B). To calculate this weuse the conditional probability formulaP r(A ∩ B)P r(A| B) =P r(B)P r(Have disease and test positive)P r(A| B) =P r(test positive)P r(True positive)P r(A| B) =P r(Total positive)P r(A| B) = 0.000090.00109P r(A| B) = 0.083• Only 8.3% of those women identified as having the disease actuallydo.45
Page 1 and 2: Contents1 Course Administration 42
Page 3 and 4: 10 Regression 20210.1 Introduction
Page 5 and 6: Resource PageDepartment of Mathemat
Page 9 and 10: 3 Data and Study Designs3.1 Basic d
Page 11 and 12: Parameter• Parameter: Fixed numbe
Page 13 and 14: TypesDiscrete - can put in one-to-o
Page 15 and 16: - How do geologists know?- Sediment
Page 17 and 18: Source: skyscrapercity.comSampling
Page 19: - 1 in 3.8 million - 3.8 million di
Page 23 and 24: Important designs• Completely ran
Page 25 and 26: B. Randomised control - of all chil
Page 27 and 28: - Reason why observational studies
Page 29 and 30: Things that go wrong• Even the be
Page 31 and 32: 4 ProbabilityFred’s DayFred awoke
Page 33 and 34: • The event that the mouse we tra
Page 35 and 36: Blood donor example - Multiplicatio
Page 37 and 38: Fair Die Example• A fair die is t
Page 39 and 40: Tree diagram rules• Add Verticall
Page 41 and 42: Tree Diagrams - Independent Stages0
Page 43: Dependent Stages• Andrew, John an
Page 47 and 48: Calculating Probabilities• Estima
Page 49 and 50: 4.3 Random VariablesRandom Variable
Page 51 and 52: Calculating the Variance• The sam
Page 53 and 54: Examples• Consider a data set XX
Page 55 and 56: Combining 2 Random Variables• If
Page 57 and 58: Fred’s DayShould Fred be worried
Page 59 and 60: Now using the complementary event r
Page 61 and 62: Mean of Binary DistributionThe mean
Page 63 and 64: • Mean number of successes• Var
Page 65 and 66: Using the Formulan = 3, x = 2, π =
Page 67 and 68: • A probability less than 0.05 is
Page 69 and 70: Endangered bird egg example solutio
Page 71 and 72: RELATIVE FREQUENCY HISTOGRAMNormal
Page 73 and 74: Finding Areas Under the Curve• In
Page 75 and 76: • Find P r(−1 < Z < 1.64)pnorm(
Page 77 and 78: INVERSE PROBLEMS USING R-COMMANDER
Page 79 and 80: CALCULATING PROBABILITIESAssume tha
Page 81 and 82: CONTINUITY CORRECTION• Normal pro
Page 83 and 84: Sample size = 10Consider the situat
Page 85 and 86: ConclusionWhat conclusion would you
Page 87 and 88: 10%5.5 6.0 6.5 7.0 7.5 8.0 8.5XWe k
Page 89 and 90: 6 Sampling Distributions and Estima
Page 91 and 92: DERIVATION IWe can think of the sam
Page 93 and 94: Solution IIPr(X > 172) = 1 - pnorm(
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qnorm(0.975) as 2.5% of the area is
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6.2.1 Sample Size CalculationEXAMPL
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The 99% Confidence IntervalThe 99%
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6.3 Comparing Two SamplesCOMPARING
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THE POOLED VARIANCEIf the variances
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SOLUTION - Calculating the confiden
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Confidence Interval for raw dataTo
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= 6.6 ± 24.6That is, -18.0 < µ in
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STANDARD DEVIATIONIf P = X n : σP
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6.4.2 Sample Size CalculationEXAMPL
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Fred for MayorFred has decided to t
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Fred’s mate Bob pointed out that
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ALTERNATIVE HYPOTHESESTwo types:Stu
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• If we use the sample standard d
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Calculate the test statistic:Test s
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Finding the pooled variance:Recall
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Caluclate the estimated standard er
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7.6 Significance and Conclusiveness
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Conclusion: There is no evidence th
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Some practical pointers• Aim for
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Fred’s Parking & Extracurricular
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Fred calculated the p-value using t
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Fred told Carol he would use odds r
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Flu vaccine example• There were 1
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8.3 Attributable Risk (AR)The attri
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• Therefore w x ≈ww+x and y z
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Aspirin study example - gastrointes
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Mobile phone example• Human expos
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OR = 0.81 with CI (0.39,1.70)• p
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• We calculate these expected cou
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• We can calculate the p value us
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χ 2 =(100 − 108.99)2 (107 − 11
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• The reason for the discrepancy
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Summary - Confidence IntervalsFacto
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Since 1 is excluded from this 95% i
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9 ANOVAFred’s Public ImageSince F
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• Previously used two sample t-te
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• This tells us if the observed d
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RESIDUAL MEAN SQUARE (s 2 e)Group A
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9.2 Post ANOVA AnalysisPOST ANOVA A
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• Hence no need to additionally c
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Since zero is excluded, and the int
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SUM OF SQUARESTotal SS = 93 2 + 100
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General Level SS = 12 x ( 9812 )2=
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ANOVA TABLEA one factor analysis of
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Data > Manage variables in active d
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DATA COMPONENTSEach data value can
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ANOVA TABLE - Two Factor FactorialS
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A B C D EChristchurch 120.5 119 122
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DataFERTILISER LEVELSEED TYPE Low M
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Interaction PlotsHypothesis test fo
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Fred’s WeightUsing the informatio
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Fred wanted to address the followin
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10 RegressionFred’s Dog/BeardFred
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2. studies which have measured bina
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• What if we colour code the poin
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Statistically• We use slightly di
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Blood pressure exampleStress(x) Blo
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Focussed look on the limits of our
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Height example• We will perform a
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Normality Assumption FAILPP PlotExp
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10.3 Confidence Intervals and Regre
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0.236 < β 1 < 0.496• We can conc
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95% confidence and prediction inter
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100m World Record Progression●●
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Negative correlation• The denomin
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Stress exampleStress(x) Blood Press
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10.5 Multiple regression• Simple
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• Appears that total lung capacit
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Multiple linear regression predicti
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Three variable modelIntepreting Mod
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• So for men we have:y = −7.066
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• Is the variable age an importan
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Extra sum of squares principle• T
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Diagnostics• Remember our Assumpt
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TableTreatmentControlBP(Y) AGE(X) B
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• So for people in the control gr
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Discussion• The value of d is the
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Multiple Linear RegressionTreatment
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• Logistic regression is sometime
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• Given the s.e.(OR) = 0.751. Thi
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• We get the same as when we calc
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Female(0)Male(1)Born Left(1) Right(
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Fred’s Dog/BeardTo help solve his
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This is a very high r 2 value and i
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Attractiveness = 4.85 − 0.03 × (
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• Because the calculator does div
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• Install R Commander from the R
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- In large random samples (n 1 and
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Course Notes - Department of Mathematics and Statistics

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