CONTENTS - Department of Mathematics and Statistics - University ...

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Example: What are the serum cholesterol levels of New Zealanders Method: Select a subgroup (sample) of people and measure their serum cholesterol levels Random sampling • choose the sample in such a way that every individual in the population has a known chance of being selected • in a simple random sample, everyone has an equal chance of being chosen • this method is the best way of obtaining a sample which is representative of the population Suppose we want to estimate mean cholesterol in the population: 15 Section 1
Sample average = true mean + error unknown random error: systematic error random error • due to natural biological variability • increasing the sample size will reduce the random fluctuations in the sample mean systematic error (=bias) • due to aspects of the design or conduct of the study which systematically distort the results • occurs if a sample is not representative of the population • cannot be reduced by increasing the sample size 16 Section 1
Page 1 and 2: CONTENTS Introduction, General Info
Page 3 and 4: SECTION 9 This section introduces t
Page 5 and 6: Support Classes There is also a Wed
Page 7 and 8: Email Contact with Students From ti
Page 9 and 10: What is the best strategy for treat
Page 11 and 12: Examples • We use the frequency w
Page 13 and 14: In some experiments the investigato
Page 15 and 16: These studies are most often descri
Page 17 and 18: • Human studies: - anatomy and ph
Page 19 and 20: Biostatistics and research: an over
Page 21 and 22: Goal of health sciences professions
Page 23 and 24: Research A process for providing an
Page 25 and 26: Does early childhood circumcision r
Page 27 and 28: Steps in the research process: Deve
Page 29 and 30: Data Analysis and Computer Software
Page 31: 1. Descriptive studies Aim: to desc
Page 35 and 36: Example: Options for studying the r
Page 37 and 38: Example RCT: LIPID study (NEJM, 199
Page 39 and 40: Example of cohort study: Study to i
Page 41 and 42: Example of case-control study Study
Page 43 and 44: 3. Summary Classification of resear
Page 45 and 46: Types of data and graphical summari
Page 47 and 48: In these examples the data are said
Page 49 and 50: 3. Rates, Ratios and Proportions Th
Page 51 and 52: Two treatments may be assessed from
Page 53 and 54: Example for Continuous Data: In a h
Page 55 and 56: Histograms: These are simple pictur
Page 57 and 58: (5) The relative (or percentage) fr
Page 59 and 60: Note The mean need not be one of th
Page 61 and 62: 0 Median Mean The mean may be unsui
Page 63 and 64: If data are highly variable there a
Page 65 and 66: Example: Find the standard deviatio
Page 67 and 68: Box-and-whisker plot This is a seco
Page 69 and 70: Example: Thirty-two traps were plac
Page 71 and 72: Interpreting Box whisker plots (Ref
Page 73 and 74: How to make the call. This is based
Page 75 and 76: [C] Measures of Disease Frequency A
Page 77 and 78: Example: In a survey of eye disease
Page 79 and 80: 2.1 Cumulative incidence is the pro
Page 81 and 82: • = Initiation of follow-up × =
Page 83 and 84:
Example In (a)-(c) calculate a rele
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Relationship between prevalence and
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HIV/AIDS Many with HIV will live fo
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[D] Measures of disease association
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1. Relative effect = Relative Risk
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Example: A randomised trial of the
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Example: Data from a cohort study o
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Note Relative risks • provide inf
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SECTION 3 This section covers a bri
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A probability is therefore like a r
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An event is “more chemotherapy pa
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Pr(O) × Pr(O) × Pr(O) = 0.47 × 0
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• Event A and B denoted by A∩ B
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Independence The idea behind the te
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A tree diagram is useful for helpin
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The tree diagram shows all possible
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In probability notation we get: A i
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Pr(A) = 0.0001 (disease prevalence)
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1. Pr(B|A) is called the sensitivit
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Notice how the probability of heart
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WHO SARS Confirmed Result Yes No To
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“Have you ever had a sexually tra
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Probability Distribution and Random
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Describing Probability Distribution
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Example: A person infected with a d
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Note: Properties 3 and 4 tell us th
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Note: These results can be extended
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SECTION 4 This section introduces b
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Now suppose that we take a sample o
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Notes 1. If these conditions are me
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(a) Find the probability distributi
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or the teacher is at fault and more
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Solution (a) X is binomial with n =
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Example (Revision) The standard dru
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The Normal Distribution This distri
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μ Y Y = f(X) X The equation of suc
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Normal distribution calculations. T
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4. Pr(-1 < Z < 1.64) = Pr(0 < Z < 1
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Some calculations 1. Pr( μ X - σ
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(c) Find the diastolic blood pressu
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∴ Pr( X = x) ⎛ 1 1 ( x − 2)
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More on the normal and Statistical
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Example: It is claimed cancer tumor
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Transforming Data If data being ana
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Now suppose we want the range of va
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REVIEW EXERCISES 4. For the standar
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SECTION 5 This section defines samp
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Normal Underlying Population Contin
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The Distribution of Sample Means Su
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Distributions of means are for samp
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The addition rule for the variance
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Example: Suppose a population has v
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A Confidence Interval for the Mean
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Therefore, we are 95% confident tha
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Example: A pharmacologist is invest
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Use of t-table when σ X is unknown
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Student’s t distribution ν -t ν
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Example: Tablets must be produced w
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100 Confidence Intervals (95%) Samp
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185 Section 5
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187 Section 5
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Example: It is claimed that males c
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Case 2: Comparing means when sample
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Example 3: Following data are 24 ho
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Note: This confidence interval tell
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Find a 95% confidence interval for
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Example: Weight loss programme agai
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Confidence Intervals for a Proporti
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small sample, the normal distributi
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0.5(1 − 0.5) 1.96 < 0.03 n 2 (1.9
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Solution: The proportions improved
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SOLUTIONS 2. (a) X is a normal dist
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[A] DISTRIBUTION SUMMARY 1. Binomia
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[B] SUMMARY: CONFIDENCE INTERVALS s
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SECTION 6 This section reviews hypo
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If the data suggest harm we are lik
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Examples Hypothesis Testing Exercis
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Notes: 1. R-cmdr and other statisti
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Step(3): Sample gives p = 27/293 =
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Step(3): Sample gives x − x = 3.4
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227 Section 6
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Notes on Hypothesis Testing 1. Ther
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(b) H 0 is not true, and the observ
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(b) Mean Diff = 36. Confidence inte
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Summary of previous results 0 = nul
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Example A clinical trial is set up
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Errors in Hypothesis Testing The le
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Some Computer packages (Minitab is
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A printout is as follows: Power and
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(c) Halve the value of sigma to 1.0
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Revision Examples 1. Exam 2006: In
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(c) (d) (e) (2 marks) A mean temper
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2. Exam 2005 An ecologist must dete
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(c) (2 marks) Find the probability
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SECTION 7 One factor analysis of va
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Example A general surgeon believes
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We now investigate the problem of h
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Estimation of Components (for refer
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There are 20 data values altogether
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υ 2 υ 1 1 2 3 4 … 30 1 15 3.05
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If n 1 = n 2 = … = n k = n (say)
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[ 2 2 2 2 2 ] s + s + s + s s 2 1 s
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Total SS = 4 2 + 12 2 + … + 89 2
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POST ANALYSIS OF VARIANCE RESULTS I
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Solution: x = 110/4 = 27.5 x = 285/
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The next two residual plots confirm
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SECTION 8 This section covers the a
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Examples of research questions that
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Example 1: What is the prevalence o
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Example 3: Is there an association
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Prospective Studies • groups are
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Confidence interval for relative ri
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Confidence interval for attributabl
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Case control studies • a group of
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a/ b The odds ratio, OR = c/ d = 32
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so a a c c ≈ and ≈ b a+ b d c+
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Chi Square Test for Contingency Tab
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We wish to know whether the data in
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Using this formula, expected number
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Example: For the drug responses, χ
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• the p-value gives the probabili
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Example 5 Is there an association b
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χ 2 2 (100−108.99) (107−111.01
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This page for reference only Income
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(a) OR = 1.90 with CI (1.23, 2.92)
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0 1 1.5 2 3 3.5 a x b x c d e x x x
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with υ = 1 degree of freedom. Sinc
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COMBINED ADMIT DECLINE Male 490 (44
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SOLUTIONS 29 1. (b) Risk (aspirin g
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SECTION 9 This section introduces t
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Regression methods are used to intr
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BAC X X X • • • • X X X X X
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FEV + + + + + + + + + + + + + + + +
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80 Y X 70 60 X X X 50 X X 40 X 100
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The method of least squares finds t
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N.B. 1. In this situation we have r
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important as they represent the err
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To illustrate, the example has x =
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That is, the total variation is par
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The F-distribution (Table in Append
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5. For the validity of the F-test,
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2. The divisor is (n - 2) rather th
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x i y i ( y − yˆ ) i i ( y − 1
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Stress (X) 55 94 64 73 96 86 Blood
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Confidence Interval for Prediction
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Y Prediction Interval X EXAMPLE: 20
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(c) (3 marks) What is the predicted
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(b) Estimated standard error = 3.47
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The denominator in formula for r is
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2 ( y i − y) ( x − x)( y − y)
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2. r = -1 is smallest value which i
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Some examples on correlation and as
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REVIEW EXERCISES 1. Physical fitnes
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SECTION 10 Multiple regression mode
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• The type of multiple regression
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The multiple linear regression mode
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Example For lung transplantation it
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It appears total lung capacity incr
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Step 3: Fit (in R-cmdr) Multiple Li
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The fitted equation is therefore ŷ
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It is 6.98 - 5.20 = 1.78 litres whe
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Are all three variables needed in t
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Test of the hypothesis H 0 : β 1 =
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95% confidence interval for coeffic
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Checking the fit of the model We do
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The points in the normal P-P plot l
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Analysis of Covariance This analysi
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At this stage the ages have been ig
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[C] Third, a regression analysis of
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The confidence interval here is the
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Notes: 1. 2 ˆβ (the coefficient o
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The logistic regression model is:
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(1) Investigating the relationship
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BUT what about the potential confou
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SECTION 11 Study design principles,
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Critical appraisal Critical apprais
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1. Introduction to critical apprais
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2. Design and appraisal of Descript
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Recall Suppose we want to estimate
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Selection bias • systematic error
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Example Life in New Zealand Survey,
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Internal validity Bias Selection bi
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Study design and critical appraisal
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Types of design: • experimental (
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Example Does smoking cause coronary
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Assessing internal validity: Bias S
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Evaluation and control of bias •
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Example of confounding: A study was
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Positive and Negative Confounding P
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Some comments on confounding: AGE a
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(3) Control of confounding during t
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Basic structure of a RCT • popula
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The way in which patients are alloc
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Randomised controlled trials : Exam
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Key results 849 randomised placebo
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Internal validity Chance • 95% co
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Intention-to-treat analysis “once
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Study design and critical appraisal
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Prospective cohort study (concurren
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Selection of participants • 5,766
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Bias Selection bias • because the
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Case-control studies Ref: “Case-C
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Selection of participants Study pop
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Key results Response rates: Cases:
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In this study: • the odds ratios
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Information bias Things done to min
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Generalisability Could we apply the
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Appendix One: The Basics This appen
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Example 3 If Z = X − μ calculate
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There are no hard and fast rules co
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Converting Fractions to Decimals Th
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X − μ Example 1: If Z = , calcul
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7. Sigma means Add Up The Greek let
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Section 2: Basic Statistical Concep
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5. Quartiles and Interquartile Rang
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9. If 1.96 X − 2.8 = , calculate
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• Probability of B given A: Pr( A
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Summary of Formulae 1. Normal Distr
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