Course Notes - Department of Mathematics and Statistics

More documents

Recommendations

Info

χ 2 =(100 − 108.99)2 (107 − 111.01)2 (111 − 108.99)2+ +108.99111.01108.99(122 − 111.01)2 (115 − 106.01)2 (112 − 107.99)2+ + +111.01106.01107.99(104 − 106.01)2 (97 − 107.99)2+ +106.01 107.99= 4.1• In this case ν = 1 × 3• There is no evidence to reject the null hypothesis. So we acceptthat there is no association between income level and cardiovasculardisease.8.6.1 Simpson’s ParadoxSimpson’s paradox - A problem when combining contingency tables• A University has a Law School and a Medical Sciences school withmen and women being admitted or declined as follows:Admit Decline TotalMale 490 210 700Female 280 220 500770 430 1200• Is there gender bias concerning admission?Calculate the Expected valuesAdmit Decline Total700(770)Male1200= 449.2 [250.8] 700Female [320.8] [179.2] 500770 430 1200• χ 2 = (490−449.2)2449.2+ . . . + . . . + (220−179.2)2179.2= 24.82• With ν = 1158
Admit Decline TotalMale 490 [449.2] 210 [250.8] 700Female 280 [320.8] 220 [179.2] 500770 430 1200Inspect values• There appears to be more men admitted, and less women admittedthan expected.What happens if we spilt admissions by school?• For Law school χ 2 = 10.38• For Medical school χ 2 = 20.45• So for both schools there is still an association between genderand admissionLet’s look at the valuesLaw SchoolAdmit Decline TotalMale 480 [495] 120 [105] 600Female 180 [165] 20 [35] 200660 140 800Medical SchoolAdmit Decline TotalMale 10 [27.5] 90 [72.5] 100Female 100 [82.5] 200 [217.5] 300110 290 400What?!• We reach the opposite conclusion to that when the schools arecombined.• So is there discrimination against men or women?• This is know as Simpson’s Paradox.159
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 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 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

Create successful ePaper yourself

Delete template?

Save as template?