Course Notes - Department of Mathematics and Statistics

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Fred’s Dog/BeardTo help solve his problem about what size dog to get, Fred summarizedthe information he got from his friends in the table below.Dog Weight (kg) Amount of dog food (kg)2 0.16655 0.3338 110 0.75020 140 1.62560 2.2580 2.5100 3120 3.5He also calculated some standard statistics used in regression.• ȳ = 44.5• ¯x = 1.6125n∑• (x i − ¯x) 2 = 11.9043i=1n∑• (y i − ȳ) 2 = 16790.5i=1n∑• (x i − ¯x)(y i − ȳ) = 439.9578i=1n∑ √• (xi − ¯x) 2 (y i − ȳ) 2 = 447.0782i=1While regression can be done simply on statistical software, it is possibleto calculate simple linear regression by hand. Since Fred did not haveaccess to his computer he decided to calculate the equation by hand.262
Next Fred calculated the β 1 :n∑(x i − ¯x)(y i − ȳ)ˆβ 1 =Next Fred calculated the β 0 :i=1n∑(x i − ¯x) 2i=1ˆβ 1 = 439.957811.9043ˆβ 1 = 36.9579ˆβ 0 = ȳ − ˆβ 1¯xˆβ 0 = 44.5 − 36.9579 × 1.6125ˆβ 0 = −15.0946Putting this information together Fred gets the following regressionequation.ŷ = −15.0946 + 36.9579xThis means that for every kilo of dog food you have per day, the sizeof the dog you can feed increases by 36 kilos (or equivalently for everyextra 100g of dog food you have per day, the size of the dog you canfeed increases by 3.6 kilos). To test whether or not this is a good model,Fred decided to calculate the r 2 value.⎛r 2 =⎜⎝⎞n∑(x i − ¯x)(y i − ȳ)i=1n∑ √ ⎟(xi − ¯x) 2 (y i − ȳ) 2 ⎠2r 2 =i=1( 439.9578447.0782r 2 = 0.9684) 2263
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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
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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
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 254 and 255: • Logistic regression is sometime
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 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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