Course Notes - Department of Mathematics and Statistics

More documents

Recommendations

Info

Polio vaccine trial• Acute viral disease spread by person-to-person contactPolio cases (deaths in red) in NZCases0 200 400 600 800 1000 12001920 1940 1960 1980 2000YearPolio vaccine trial• Mass vaccination of children in NZ in 1961 and 1962• Followed US trials involving 1.8M children- Trial needed because of earlier failures- Massive size needed because of low prevalence• Controversy over the design – parental consent required and soonly studying volunteers.Polio vaccine trial II• Two designs usedA. Observed control - 1st and 3rd graders acted as controls and2nd graders with parental consent vaccinated.24
B. Randomised control - of all children who had parental consent,half were randomly allocated to the control group and theother half vaccinated.- The control group were “vaccinated” with a saline solution- Children, Drs, researchers didn’t know which child was inwhich group until after the experiment.Polio vaccine trial IIIDesign Trt group n Cases Rate (per 100,000)Observed Control Vaccinated 221998 56 25.2Observed Control Controls 725173 391 53.9Randomised Control Vaccinated 200745 57 28.4Randomised Control Placebo 201229 142 70.6• Observed control design biased against the vaccine.- Confounding of vaccine effect and ‘parental consent’ effect.- Children from poorer families had a natural level of immunityand less likely to receive consent for involvement in the trial.Chance effect or real?• Just by chance things almost always differ. Could this be chance?• Suppose we have two unfair coins, a ‘Control’ coin and a ‘Vaccine’coin• We toss the control coin 201,229 times and the other one 200,645times.• If they had the same chance of coming up heads what is theprobability of observing the outcome 142 heads for the controlcoin and 57 for the vaccine coin, or a more extreme one?- More than one billion to one against- Coins are not the same.25
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: Important designs• Completely ran
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 and 44: Dependent Stages• Andrew, John an
Page 45 and 46: 0.90BBiopsy +ve (true positive)0.00
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(
Page 95 and 96:
qnorm(0.975) as 2.5% of the area is
Page 97 and 98:
6.2.1 Sample Size CalculationEXAMPL
Page 99 and 100:
The 99% Confidence IntervalThe 99%
Page 101 and 102:
6.3 Comparing Two SamplesCOMPARING
Page 103 and 104:
THE POOLED VARIANCEIf the variances
Page 105 and 106:
SOLUTION - Calculating the confiden
Page 107 and 108:
Confidence Interval for raw dataTo
Page 109 and 110:
= 6.6 ± 24.6That is, -18.0 < µ in
Page 111 and 112:
STANDARD DEVIATIONIf P = X n : σP
Page 113 and 114:
6.4.2 Sample Size CalculationEXAMPL
Page 115 and 116:
Fred for MayorFred has decided to t
Page 118 and 119:
Fred’s mate Bob pointed out that
Page 120 and 121:
ALTERNATIVE HYPOTHESESTwo types:Stu
Page 122 and 123:
• If we use the sample standard d
Page 124 and 125:
Calculate the test statistic:Test s
Page 126 and 127:
Finding the pooled variance:Recall
Page 128 and 129:
Caluclate the estimated standard er
Page 130 and 131:
7.6 Significance and Conclusiveness
Page 132 and 133:
Conclusion: There is no evidence th
Page 134 and 135:
Some practical pointers• Aim for
Page 136 and 137:
Fred’s Parking & Extracurricular
Page 138 and 139:
Fred calculated the p-value using t
Page 140 and 141:
Fred told Carol he would use odds r
Page 142 and 143:
Flu vaccine example• There were 1
Page 144 and 145:
8.3 Attributable Risk (AR)The attri
Page 146 and 147:
• Therefore w x ≈ww+x and y z
Page 148 and 149:
Aspirin study example - gastrointes
Page 150 and 151:
Mobile phone example• Human expos
Page 152 and 153:
OR = 0.81 with CI (0.39,1.70)• p
Page 154 and 155:
• We calculate these expected cou
Page 156 and 157:
• We can calculate the p value us
Page 158 and 159:
χ 2 =(100 − 108.99)2 (107 − 11
Page 160 and 161:
• The reason for the discrepancy
Page 162 and 163:
Summary - Confidence IntervalsFacto
Page 164 and 165:
Since 1 is excluded from this 95% i
Page 166 and 167:
9 ANOVAFred’s Public ImageSince F
Page 168 and 169:
• Previously used two sample t-te
Page 170 and 171:
• This tells us if the observed d
Page 172 and 173:
RESIDUAL MEAN SQUARE (s 2 e)Group A
Page 174 and 175:
9.2 Post ANOVA AnalysisPOST ANOVA A
Page 176 and 177:
• Hence no need to additionally c
Page 178 and 179:
Since zero is excluded, and the int
Page 180 and 181:
SUM OF SQUARESTotal SS = 93 2 + 100
Page 182 and 183:
General Level SS = 12 x ( 9812 )2=
Page 184 and 185:
ANOVA TABLEA one factor analysis of
Page 186 and 187:
Data > Manage variables in active d
Page 188 and 189:
DATA COMPONENTSEach data value can
Page 190 and 191:
ANOVA TABLE - Two Factor FactorialS
Page 192 and 193:
A B C D EChristchurch 120.5 119 122
Page 194 and 195:
DataFERTILISER LEVELSEED TYPE Low M
Page 196 and 197:
Interaction PlotsHypothesis test fo
Page 198 and 199:
Fred’s WeightUsing the informatio
Page 200 and 201:
Fred wanted to address the followin
Page 202 and 203:
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 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 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
show all

Course Notes - Department of Mathematics and Statistics

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?