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or the teacher is at fault and more pupils develop OOS. This second alternative is usually taken and therefore we conclude the teacher has a higher incidence of OOS. Notes 1. It is the size of the probability of this observed “event” or a more extreme and convincing event which results in the conclusion (more than 4). 2. 0.0196 is a chance of just under 2 per 100 (2%). 3. A probability less than 0.05 is (by convention) taken to imply an event is rare or unlikely to occur. 4. A probability above 0.05 often means an event is not unusual. If the violin teacher had produced such a probability then the teaching would not be at all unusual in relation to incidence of OOS. 129 Section 4
Binomial Examples and Normal Distribution Example: (artificial data) A sociological report suggests that 75% of Maori children under 18 live with both parents. A random sample of 20 Maori children is selected, and X is the binomial random variable for the number of these 20 who live with both parents. (a) Define the parameters of the distribution of X. (b) Find Pr(X = 15). (c) Find the probability that 11 or fewer live with both parents (i.e. Pr(X ≤ 11)). (d) A random sample of 20 New Zealand Caucasian children had only 11 living with both parents. Does this result provide any evidence to support the claim that 75% of NZ Caucasian children live with both parents 130 Section 4
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CONTENTS Introduction, General Info
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SECTION 9 This section introduces t
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Support Classes There is also a Wed
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Email Contact with Students From ti
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What is the best strategy for treat
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Examples • We use the frequency w
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In some experiments the investigato
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These studies are most often descri
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• Human studies: - anatomy and ph
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Biostatistics and research: an over
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Goal of health sciences professions
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Research A process for providing an
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Does early childhood circumcision r
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Steps in the research process: Deve
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Data Analysis and Computer Software
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1. Descriptive studies Aim: to desc
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Sample average = true mean + error
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Example: Options for studying the r
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Example RCT: LIPID study (NEJM, 199
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Example of cohort study: Study to i
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Example of case-control study Study
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3. Summary Classification of resear
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Types of data and graphical summari
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In these examples the data are said
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3. Rates, Ratios and Proportions Th
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Two treatments may be assessed from
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Example for Continuous Data: In a h
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Histograms: These are simple pictur
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(5) The relative (or percentage) fr
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Note The mean need not be one of th
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0 Median Mean The mean may be unsui
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If data are highly variable there a
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Example: Find the standard deviatio
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Box-and-whisker plot This is a seco
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Example: Thirty-two traps were plac
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Interpreting Box whisker plots (Ref
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How to make the call. This is based
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[C] Measures of Disease Frequency A
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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
Page 95 and 96: Example: Data from a cohort study o
Page 97 and 98: Note Relative risks • provide inf
Page 100 and 101: SECTION 3 This section covers a bri
Page 102 and 103: A probability is therefore like a r
Page 104 and 105: An event is “more chemotherapy pa
Page 106 and 107: Pr(O) × Pr(O) × Pr(O) = 0.47 × 0
Page 108 and 109: • Event A and B denoted by A∩ B
Page 110 and 111: Independence The idea behind the te
Page 112 and 113: A tree diagram is useful for helpin
Page 114 and 115: The tree diagram shows all possible
Page 116 and 117: In probability notation we get: A i
Page 118 and 119: Pr(A) = 0.0001 (disease prevalence)
Page 120 and 121: 1. Pr(B|A) is called the sensitivit
Page 122 and 123: Notice how the probability of heart
Page 124 and 125: WHO SARS Confirmed Result Yes No To
Page 126 and 127: “Have you ever had a sexually tra
Page 128 and 129: Probability Distribution and Random
Page 130 and 131: Describing Probability Distribution
Page 132 and 133: Example: A person infected with a d
Page 134 and 135: Note: Properties 3 and 4 tell us th
Page 136 and 137: Note: These results can be extended
Page 138 and 139: SECTION 4 This section introduces b
Page 140 and 141: Now suppose that we take a sample o
Page 142 and 143: Notes 1. If these conditions are me
Page 144 and 145: (a) Find the probability distributi
Page 148 and 149: Solution (a) X is binomial with n =
Page 150 and 151: Example (Revision) The standard dru
Page 152 and 153: The Normal Distribution This distri
Page 154 and 155: μ Y Y = f(X) X The equation of suc
Page 156 and 157: Normal distribution calculations. T
Page 158 and 159: 4. Pr(-1 < Z < 1.64) = Pr(0 < Z < 1
Page 160 and 161: Some calculations 1. Pr( μ X - σ
Page 162 and 163: (c) Find the diastolic blood pressu
Page 164 and 165: ∴ Pr( X = x) ⎛ 1 1 ( x − 2)
Page 166 and 167: More on the normal and Statistical
Page 168 and 169: Example: It is claimed cancer tumor
Page 170 and 171: Transforming Data If data being ana
Page 172 and 173: Now suppose we want the range of va
Page 174 and 175: REVIEW EXERCISES 4. For the standar
Page 176 and 177: SECTION 5 This section defines samp
Page 178 and 179: Normal Underlying Population Contin
Page 180 and 181: The Distribution of Sample Means Su
Page 182 and 183: Distributions of means are for samp
Page 184 and 185: The addition rule for the variance
Page 186 and 187: Example: Suppose a population has v
Page 188 and 189: A Confidence Interval for the Mean
Page 190 and 191: Therefore, we are 95% confident tha
Page 192 and 193: Example: A pharmacologist is invest
Page 194 and 195: 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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