Course Notes - Department of Mathematics and Statistics

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3 Data and Study Designs3.1 Basic definitionsDefinitions• Each question has a numerical answer (counts, probabilities, proportions,...).• Need a toolkit for numerical description and for answering numericalquestions.• Inference – formal name given to learning from data using statisticaltools.• We need some agreed terminology.Population• Population Complete set of entities or elements or units or subjectsthat we wish to describe or make inference about. Real orhypothetical.Well-defined- The collection of words in poems by W. B. Yeats.- All the rimu trees in Tongariro National Park.Not well-defined- The population of New Zealand. Right now? Past? Future?- Banks Peninsula Hector’s dolphins. Which dolphins shouldwe include?- Target population in a drug trial. All alive or who will ever beborn? Over a certain age? Only those people who can affordthe drug?Which population have we studied? Which are we interested in?9
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 10 and 11: Sample• A Sample is a subset of a
Page 12 and 13: 2h 15:25 (2003)2h 03:38 (2011)- An
Page 14 and 15: • Think of it as a metaphor - if
Page 16 and 17: Major sources of data• Sample sur
Page 18 and 19: Gamebird survey results4 May 25 May
Page 22 and 23: Replication and randomisation• Re
Page 24 and 25: Polio vaccine trial• Acute viral
Page 26 and 27: Confounding• Between 1972 and 197
Page 28 and 29: Case-Control Study• Retrospective
Page 30 and 31: Good and Bad studies• Reliable sa
Page 32 and 33: • Bayesians consider probability
Page 34 and 35: • In this case the addition rule
Page 36 and 37: Hospital Patients - Question 2• A
Page 38 and 39: • Or we can calculate P r(A ∩ B
Page 40 and 41: Handedness vs Gender - Combined Pro
Page 42 and 43: P r(X = 1) = 1 − 0.288 − 0.216
Page 44 and 45: Definitions IIPositive Predictive V
Page 46 and 47: Negative Predictive value• Find t
Page 48 and 49: Sensitive Survey Questions• Somet
Page 50 and 51: Probability DistributionNote that f
Page 52 and 53: X = x i Pr(X = x i )0 0.101 0.252 0
Page 54 and 55: Rearranging the FormulaC = 5 (F −
Page 56 and 57: • The variance of Z can be calcul
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Pr(D ∩ T)Pr(D| T) =Pr(T)P r(D| T)
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5 Probability DistributionsFred’s
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Since all the Y’s come from the s
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PROBABILITY OF X SUCCESSESThe proba
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2.( nP r (X = x) = πk)x (1 − π)
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Use R commander or R excel4. In a p
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5.2 Normal DistributionDISTRIBUTION
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Normal Distribution Notes• The gr
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• Find P r(Z > 1.64)pnorm(1.64,lo
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• Find P r(−2 < Z < 2)pnorm(2)-
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AREAS UNDER THE CURVE USING R-COMMA
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Find the height which is exceeded b
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WHEN IS THE USE OF A NORMAL APPROXI
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Here the mean is n × π = 100 × 0
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drug is more effective than the sta
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What is the probability a player co
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6.1 Introduction to Sampling Distri
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• If sample size n is greater, th
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6.2 Confidence Interval for the Mea
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EXAMPLEA pharmacologist is investig
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6.2.2 The t DistributionTHE t DISTR
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6.2.3 Interpreting a Confidence Int
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• Standard error of the differenc
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[2.] Comparing means for small samp
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6.3.2 Transforming DataTRANSFORMING
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6.3.3 Comparing Two Non-Independent
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CALCULATING THE CONFIDENCE INTERVAL
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SolutionThe 95% C.I. for π is:√p
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6.4.3 Confidence Interval for Diffe
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As Fred wants to know the proportio
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7 Hypothesis TestingFred’s Parkin
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• If the standard deviation is un
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Find the p-value:Pr(Z < -2.78)=0.00
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EXAMPLE THREE (B) - HYPOTHESIS TEST
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7.4 Hypothesis Test Difference Two
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7.5 Interpreting the p-valueINTERPR
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(a) Mean difference = 40, confidenc
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INCONCLUSIVE CONFIDENCE INTERVALSNo
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ERROR TYPESAccept RejectH 0 True Co
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Part IIFred sets the hypothesis out
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8 Contingency TablesFred’s Garden
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• For the purpose of this course
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What can we do with this informatio
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So in our swimming example:• Odds
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Confidence interval for relative ri
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So the confidence interval is given
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ln(OR) ± 1.96 × s.e.(ln(OR))−0.
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8.6 Chi Square Test for Contingency
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• We can continue to calculate th
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Some notes on χ 2• Maximum power
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Admit Decline TotalMale 490 [449.2]
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• ¯x = Σn ix iN = 2174868 = 2.5
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Fred’s GardenSo Fred recorded the
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Since 1 is excluded there is strong
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AdoptiveBiological Parents’ SESPa
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EXAMPLE: Cuckoo Egg LengthsCompare
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9.1.3 F DistributionF DISTRIBUTION
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EXAMPLE20 children allocated random
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RESIDUAL MEAN SQUARE (s 2 e)Group A
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Notes• Degrees of freedom are 15
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9.4 Two factor ANOVATWO FACTOR ANOV
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• Conclusion:There is some eviden
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NOTES• We need to give two parts
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CONFOUNDING VARIABLEPotential varia
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INTERACTIONA significant interactio
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CALCULATING INTERACTION EFFECTInter
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• Convert numerical group names i
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• The city effect is not the same
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General Level SS = 18 x ( 26418 )2=
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Calculating the Test StatisticTest
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And finally he needed to work out t
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2. H 0 The mean IQs of children wit
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He sent her some data resulting fro
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More complex relationships• Look
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Equation for a straight line• We
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Instead use method of least squares
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• Compute interceptˆβ 0 = ȳ
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An example - Analysis of variance
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Normality Assumption P-P plotPP Plo
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2nd Residual Plot FAILStandardized
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n∑(x i − ¯x) 2 = 23485.35i=1x
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Prediction Interval• We can find
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• A confidence interval estimates
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Correlation• The correlation coef
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Non-linear Correlation• In the pr
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• Therefore r 2 = 0.8931, so 89.3
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• where ˆβ 0 , ˆβ 1 , ˆβ 2
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• Age alone.ŷ = 5.0688 + 0.0359a
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Dummy variables in lung capacity ex
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Height Vs Lung Capacity• From bef
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Three variable modelTest of the Hyp
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• Giving a confidence interval of
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Extra sum of squares modelSource of
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●●LinearModel.2res−2 −1 0 1
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• At this stage the ages have bee
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Multiple Regression• Now lets per
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Multiple Linear Regression GraphTre
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Source of variation SS DF MS FRegre
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p1−p = expβ 0 + β 1 x 1 + . . .
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• logit = −1.674 + 1.841x 1Logi
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• OR = 6.98 (1.524,27.987)• The
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Example• The adjusted OR and CI w
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Next Fred calculated the β 1 :n∑
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This gives a confidence interval of
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ATools for assignmentsCommon Mistak
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Correct Approach√• Evaluate 1.9
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BSummary of Formulae1. Normal Distr
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Estimate df (ν) Multiplier Standar
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Course Notes - Department of Mathematics and Statistics

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