Course Notes - Department of Mathematics and Statistics

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• Logistic regression is sometimes refered to as the Logit model.• In logistic regression the predictor variables can be either continuousor categorical(binary).• Like multiple regression, we can use logistic regression to:1. control for confounding2. investigate the effect of several variables on the outcome variableat one time.• Logistic regression can be used with any study type as long as ithas a binary outcome.• The logistic regression model is:logit(p) = ln( p1−p ) = β 0 + β 1 x 1 + . . . + β k x k + ɛ• Y is the binary outcome variable• p is the probability that a particular event will occur i.e. Pr(Y=1)• x 1 , x 2 , . . . , x k are the explanatory variables• β 0 is the intercept• β 1 , β 2 , . . . , β kInterpreting the model• p1−pis the odds of the event occuring.• Ln( p1−p) is the log odds or the logit.• The regression coefficient β i represents the change in the log oddsfor a 1-unit change in x i .• To calculate the intercept and slope coefficients is rather complexand involves a method known as maximum likelihood.• The details of which we will not worry about here, rather letRexcel do the work.• We can calculate the odds of an event by254
p1−p = expβ 0 + β 1 x 1 + . . . + β k x k + ɛ• ˆβ i is interpreted as the log of the odds ratio for explanatory variablex i .• We can obtain the odds ratio from ˆβ i by using formula: OR =e ˆβ i.Dieldrin example• In Western Australia it is required by law to treat all new housesfor termites. The treatment contains a chemical know as dieldrin,there is a fear that the chemical can get into mothers breast milk.There is a threshold which is acceptable (
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Contents1 Course Administration 42
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10 Regression 20210.1 Introduction
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Resource PageDepartment of Mathemat
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3 Data and Study Designs3.1 Basic d
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Parameter• Parameter: Fixed numbe
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TypesDiscrete - can put in one-to-o
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- How do geologists know?- Sediment
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Source: skyscrapercity.comSampling
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- 1 in 3.8 million - 3.8 million di
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Important designs• Completely ran
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B. Randomised control - of all chil
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- Reason why observational studies
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Things that go wrong• Even the be
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4 ProbabilityFred’s DayFred awoke
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• The event that the mouse we tra
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Blood donor example - Multiplicatio
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Fair Die Example• A fair die is t
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Tree diagram rules• Add Verticall
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Tree Diagrams - Independent Stages0
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Dependent Stages• Andrew, John an
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0.90BBiopsy +ve (true positive)0.00
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Calculating Probabilities• Estima
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4.3 Random VariablesRandom Variable
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Calculating the Variance• The sam
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Examples• Consider a data set XX
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Combining 2 Random Variables• If
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Fred’s DayShould Fred be worried
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Now using the complementary event r
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Mean of Binary DistributionThe mean
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• Mean number of successes• Var
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Using the Formulan = 3, x = 2, π =
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• A probability less than 0.05 is
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Endangered bird egg example solutio
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RELATIVE FREQUENCY HISTOGRAMNormal
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Finding Areas Under the Curve• In
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• Find P r(−1 < Z < 1.64)pnorm(
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INVERSE PROBLEMS USING R-COMMANDER
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CALCULATING PROBABILITIESAssume tha
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CONTINUITY CORRECTION• Normal pro
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Sample size = 10Consider the situat
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ConclusionWhat conclusion would you
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10%5.5 6.0 6.5 7.0 7.5 8.0 8.5XWe k
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6 Sampling Distributions and Estima
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DERIVATION IWe can think of the sam
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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
Page 204 and 205: 2. studies which have measured bina
Page 206 and 207: • What if we colour code the poin
Page 208 and 209: Statistically• We use slightly di
Page 210 and 211: Blood pressure exampleStress(x) Blo
Page 212 and 213: Focussed look on the limits of our
Page 214 and 215: Height example• We will perform a
Page 216 and 217: Normality Assumption FAILPP PlotExp
Page 218 and 219: 10.3 Confidence Intervals and Regre
Page 220 and 221: 0.236 < β 1 < 0.496• We can conc
Page 222 and 223: 95% confidence and prediction inter
Page 224 and 225: 100m World Record Progression●●
Page 226 and 227: Negative correlation• The denomin
Page 228 and 229: Stress exampleStress(x) Blood Press
Page 230 and 231: 10.5 Multiple regression• Simple
Page 232 and 233: • Appears that total lung capacit
Page 234 and 235: Multiple linear regression predicti
Page 236 and 237: Three variable modelIntepreting Mod
Page 238 and 239: • So for men we have:y = −7.066
Page 240 and 241: • Is the variable age an importan
Page 242 and 243: Extra sum of squares principle• T
Page 244 and 245: Diagnostics• Remember our Assumpt
Page 246 and 247: TableTreatmentControlBP(Y) AGE(X) B
Page 248 and 249: • So for people in the control gr
Page 250 and 251: Discussion• The value of d is the
Page 252 and 253: Multiple Linear RegressionTreatment
Page 256 and 257: • Given the s.e.(OR) = 0.751. Thi
Page 258 and 259: • We get the same as when we calc
Page 260 and 261: Female(0)Male(1)Born Left(1) Right(
Page 262 and 263: Fred’s Dog/BeardTo help solve his
Page 264 and 265: This is a very high r 2 value and i
Page 266 and 267: Attractiveness = 4.85 − 0.03 × (
Page 268 and 269: • Because the calculator does div
Page 270 and 271: • Install R Commander from the R
Page 272 and 273: - In large random samples (n 1 and
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Course Notes - Department of Mathematics and Statistics

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