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Student’s t distribution ν -t ν 0 t ν T 2p 0.100 0.050 0.020 0.010 p 0.050 0.025 0.010 0.005 1 2 3 4 5 6 7 1.895 2.365 2.998 3.500 8 9 10 120 ∝ 1.645 1.960 2.326 2.576 Area (p) or probability p refers to the area of one tail 2p gives the combined area of both tails (View the t-distribution above as a slight modification to the normal distribution Z. 179 Section 5
Notes: 1. The interval is wide when samples small. That is, less precision in estimates. 2. This last example is the most common situation where: the population is assumed to be normal; μ X and σ X are both unknown; σ X is estimated by s X from a random sample of size n. 3. Even for large n the t-table is used. Last row of table has 1.96 and 2.58 row of t. 4. From the point of view of exams we shall accept the normal distribution value for degrees of freedom greater than 30. 180 Section 5
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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
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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
Page 146 and 147: or the teacher is at fault and more
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
Page 198 and 199: Example: Tablets must be produced w
Page 200 and 201: 100 Confidence Intervals (95%) Samp
Page 202 and 203: 185 Section 5
Page 206 and 207: Example: It is claimed that males c
Page 208 and 209: Case 2: Comparing means when sample
Page 210 and 211: Example 3: Following data are 24 ho
Page 212 and 213: Note: This confidence interval tell
Page 214 and 215: Find a 95% confidence interval for
Page 216 and 217: Example: Weight loss programme agai
Page 218 and 219: Confidence Intervals for a Proporti
Page 220 and 221: small sample, the normal distributi
Page 222 and 223: 0.5(1 − 0.5) 1.96 < 0.03 n 2 (1.9
Page 224 and 225: Solution: The proportions improved
Page 226 and 227: SOLUTIONS 2. (a) X is a normal dist
Page 228 and 229: [A] DISTRIBUTION SUMMARY 1. Binomia
Page 230 and 231: [B] SUMMARY: CONFIDENCE INTERVALS s
Page 232 and 233: SECTION 6 This section reviews hypo
Page 234 and 235: If the data suggest harm we are lik
Page 236 and 237: Examples Hypothesis Testing Exercis
Page 238 and 239: Notes: 1. R-cmdr and other statisti
Page 240 and 241: Step(3): Sample gives p = 27/293 =
Page 242 and 243: Step(3): Sample gives x − x = 3.4
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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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