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

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Research questions relevant to this course: Epidemiology: the study of distribution and determinants of disease frequency Clinical research: the study of questions relating to care of patients Descriptive questions: What is the distribution of a disease What is the natural history of a disease Analytic questions: What are the causes of a disease Will this approach prevent disease Does this treatment improve outcome 11 Section 1
Data Analysis and Computer Software Easy to use software is essential for data management and data analysis. In this course R- cmdr (Statistical Package for the Social Sciences) will be used. This package is widely available on campus, used in most Departments which specify first year statistics as a pre-requisite, and widely available internationally. At school you may have used EXCEL. Possibly at University you have used EXCEL. EXCEL is excellent for data management and reporting but is poor for statistical analyses and clumsy for graphical procedures. R-cmdr is easy to use with good pull down menu options. There are three windows in R-cmdr • Data Editor (where data being analysed are located) • Output Window (where results appear) • Syntax Window (not used in this course) 12 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: Steps in the research process: Deve
Page 31 and 32: 1. Descriptive studies Aim: to desc
Page 33 and 34: Sample average = true mean + error
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
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2.1 Cumulative incidence is the pro
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• = Initiation of follow-up × =
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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
Page 518 and 519:
Summary of Formulae 1. Normal Distr
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