John%20Matthews,%20Professor%20of%20Medical%20Statistics,%20School%20of%20Mathematics%20and%20Statistics

1
Introductory Statistics

• John Matthews, Professor of Medical Statistics,
  School of Mathematics and Statistics
• Janine Gray, Senior Lecturer and Deputy Director,
  Newcastle Clinical Trials Unit

University of Newcastle-upon-Tyne
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Course Outline

• Data Description
• Mean, Median, Standard Deviation
• Graphs
• The Normal Distribution
• Populations and Samples
• Confidence intervals and p-values
• Estimation and Hypothesis testing
• Continuous data
• Categorical data
• Regression and Correlation

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Course Objectives

• To have an understanding of the Normal
  distribution and its relationship to common
  statistical analyses
• To have an understanding of basic statistical
  concepts such as confidence intervals and
  p-values
• To know which analysis is appropriate for
  different types of data

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Recommended Textbooks

• Swinscow TDV and Campbell MJ. Statistics at
  Square One (10th edn). BMJ Books
• Altman DG. Practical Statistics for Medical
  Research. Chapman and Hall
• Bland M. An Introduction to Medical Statistics.
  Oxford Medical Publications
• Campbell MJ Machin D. Medical Statistics A
  Commonsense Approach. Wiley

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Other reading

• Chinn S. Statistics for the European Respiratory
  Journal. Eur Respir J 2001 18393-401
• www.mas.ncl.ac.uk/njnsm/medfac/MDPhD/notes.htm
• BMJ statistics notes

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Types of Data

• Numerical Data
• discrete
• number of lesions
• number of visits to GP
• continuous
• height
• lesion area

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Types of Data

• Categorical
• unordered
• Pregnant/Not pregnant
• married/single/divorced/separated/widowed
• ordered (ordinal)
• minimal/moderate/severe/unbearable
• Stage of breast cancer I II III IV

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Exercise

• What type are the following variables?a) sexb)
  diastolic blood pressurec) diagnosisd) heighte)
  family sizef) cancer stage

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Types of Data

• Outcome/Dependent variable
• outcome of interest
• e.g. survival, recovery
• Explanatory/Independent variable
• treatment group
• age
• sex

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Histogram of Birthweight (grams) at 40 weeks GA
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Summary Statistics

• Location
• Mean (average value)
• Median (middle value)
• Mode (most frequently occurring value)
• Variability
• Variance/SD
• Range
• Centiles

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Birthweights (g) at 40 weeks Gestation

• mean 3441g
• median 3428g
• sd 434g
• min 2050g
• max 4975g
• range 2925g
  

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Boxplot
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Symmetric Data

• mean median (approx) ?
• standard deviation ?

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Skew Data

• median "typical" value ?
• mean affected by extreme values - larger than
  median ?
• SD fairly meaningless ?
• centiles (less affected by extreme
  values/outliers) ?

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Half of all doctors are below average.

• Even if all surgeons are equally good, about half
  will have below average results, one will have
  the worst results, and the worst results will be
  a long way below average
• Ref. BMJ 1998 3161734-1736

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Discrete Data Principal diagnosis of patients in
Tooting Bec Hospital
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Bar Chart
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Summarising data - Summary

• Choosing the appropriate summary statistics and
  graph depends upon the type of variable you have
• Categorical (unordered/ordered)
• Continuous (symmetric/skew)

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The Normal Distribution

• N(???2?
• ????unknown population mean - estimate using
  sample mean
• ????unknown population SD - estimate using sample
  SD
• Birthweight is N(3441, 4342)

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N(0,1) - Standard Normal Distribution
68 within 1
SD Units
95 within 1.96
99 within 2.58
z - SD units
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Birthweight (g) at 40 weeks
95 within 1.96 SDs 2590 - 4292 grams
99 within 2.58 SDs 2321 - 4561 grams
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Further Reading

• http//www.mas.ncl.ac.uk/njnsm/medfac/docs/intro.
  pdf
• Altman DG, Bland JM (1996) Presentation of
  numerical data. BMJ 312, 572
• Altman DG, Bland JM. (1995) The normal
  distribution. BMJ 310, 298.

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Samples and Populations

• Use samples to estimate population quantities
  (parameters) such as disease prevalence, mean
  cholesterol level etc
• Samples are not interesting in their own right -
  only to infer information about the population
  from which they are drawn
• Sampling Variation
• Populations are unique - samples are not.

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Sample and Populations

• How much might these estimates vary from sample
  to sample?
• Determine precision of estimates (how close/far
  away from the population?)

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(Artifical) example

• Have 5000 measurements of diastolic blood
  pressure from airline pilots. This accounts for
  ALL airline pilots and is the population of
  airline pilots.
• (Artificial example - if we had the whole
  population we wouldnt need to sample!!)
• Since we have the population, we know the true
  population characteristics. It is these we are
  trying to estimate from a sample.

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Population distribution of diastolic BP from
Airline Pilots (in mmHg)
True mean 78.2 True SD 9.4
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Example

• Write each measurement on a piece of paper and
  put into a hat.
• Draw 5 pieces of paper and calculate the mean of
  the BP.
• replace and repeat 49 more times
• End up with 50 (different) estimates of mean BP

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Sampling Distribution

• Each estimate of the mean will be different.
• Treat this as a random sample of means
• Plot a histogram of the means.
• This is an estimate of the sampling distribution
  of the mean.
• Can get the sampling distribution of any
  parameter in a similar way.

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Distribution of the mean
? 78.2, ? 9.4
Population
50 samples N5
50 samples N10
50 samples N100
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Distribution of the Mean

• BUT! Dont need to take multiple samples
• Standard error of the mean
• SE of the mean is the SD of the distribution of
  the sample mean

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Distribution of Sample Mean

• Distribution of sample mean is Normal regardless
  of distribution of sample(unless small or very
  skew sample)
• SOCan apply Normal theory to sample mean also

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Distribution of Sample Mean

• i.e. 95 of sample means lie within 1.96 SEs of
  (unknown) true mean
• This is the basis for a 95 confidence interval
  (CI)
• 95 CI is an interval which on 95 of occasions
  includes the population mean

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Example

• 57 measurements of FEV1 in male medical students

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Example

• 95 of population lie withini.e. within 4.06
  1.96?0.67, from 2.75 to 5.38 litres

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Example

• Thus for FEV1 data, 95 chance that the interval
  contains the true population meani.e. between
  3.89 and 4.23 litres
• This is the 95 confidence interval for the mean

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Confidence Intervals

• The confidence interval (CI) measures
  uncertainty. The 95 confidence interval is the
  range of values within which we can be 95 sure
  that the true value lies for the whole of the
  population of patients from whom the study
  patients were selected. The CI narrows as the
  number of patients on which it is based
  increases.

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Standard Deviations Standard Errors

• The SE is the SD of the sampling distribution (of
  the mean, say)
• SE SD/vN
• Use SE to describe the precision of estimates
  (for example Confidence intervals)
• Use SD to describe the variability of samples,
  populations or distributions (for example
  reference ranges)

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The t-distribution

• When N is small, estimate of SD is particularly
  unreliable and the distribution of sample mean is
  not Normal
• Distribution is more variable - longer tails
• Shape of distribution depends upon sample size
• This distribution is called the t-distribution

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N2
t(1) 95 within 12.7
N(0,1)
t(1)
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N10
N(0,1)
t(9) 95 within 2.26
t(9)
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N30
t(29) 95 within 2.04
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t-distribution

• As N becomes larger, t-distribution becomes more
  similar to Normal distribution
• Degrees of Freedom (DF)- sample size - 1
• DF measure of amount of information contained in
  data set

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Implications

• Confidence interval for the mean
• Sample size lt 30 Use t-distribution
• Sample size gt 30 Use either Normal or t
  distribution
• Note Stats packages (generally) will
  automatically use the correct distribution for
  confidence intervals

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Example

• Numbers of hours of relief obtained by 7
  arthritic patients after receiving a new drug
  2.2, 2.4, 4.9, 3.3, 2.5, 3.7, 4.3
• Mean 3.33, SD 1.03, DF 6, t(5) 2.45
• 95 CI 3.33 2.45??1.03/ ?72.38 to 4.28 hours
• Normal 95 CI 3.33 1.96??1.03/ ?72.57 to
  4.09 hours TOO NARROW!!

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Hypothesis Testing

• Enables us to measure the strength of evidence
  supplied by the data concerning a proposition of
  interest
• In a trial comparing two treatments there will
  ALWAYS be a difference between the estimates for
  each treatment - a real difference or random
  variation?

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Null Hypothesis

• Study hypothesis - hypothesis in the mind of the
  investigator (patients with diabetes have raised
  blood pressure)
• Null hypothesis is the converse of the study
  hypothesis - aim to disprove it (patients with
  diabetes do not have raised blood pressure)
• Hypothesis of no effect/difference

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Two-Sample t-test

• Two independent samples
• Can the two samples be considered to be the same
  with respect to the variable you are measuring or
  are they different?
• Sample means will ALWAYS be different - real
  difference or random variation?
• ASSUMPTION Data are normally distributed and
  SD in each group similar

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Two-Sample t-test

• 24 hour total energy expenditure (MJ/day) in
  groups of lean and obese women
• Do the women differ in their energy expenditure?
• Null hypothesis energy expenditure in lean and
  obese women is the same

50
Boxplot of energy expenditure MJ/day
51
Two-sample t-test

• Summary statistics lean obeseMean 8.1 10.3
• SD 1.2 1.4
• N 13 9
• Difference in means 10.3 - 8.1 2.2
• SE difference 0.57 (weighted average)

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Two Sample t-test

• Test statistic is 2.2/0.57 3.9
• N1 N2 - 2 DF ( 20)
• Calculate the probability of observing a value at
  least as extreme as 3.9 if the null hypothesis is
  true
• If the null hypothesis is true, the test
  statistic should have a t-distribution with 20 df
  (df N1N2-2)

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Two Sample t-test

• 95 of values from t-distribution with 20 DF lie
  between -2.09 and 2.09
• Probability of observing a value as extreme or
  more extreme than 3.9 in a t-distribution with 20
  df is 0.001
• Only a very small probability that the value of
  3.9 fits reasonably with a t-distribution with 20
  df
• Conclude that energy expenditure is significantly
  different between lean and obese women

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The P-value

• The P-value is the probability of observing a
  test statistic at least as extreme as that
  observed if the null hypothesis is true

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t distribution with 20 df
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Confidence Interval for the difference in two
means

• 95 CI 2.2 - 2.09?0.57 to 2.2 2.09?0.57
• or from 1.05 to 3.41 MJ/day
• Thus we are 95 confident that obese women use
  between 1.05 and 3.41 MJ/day energy more than
  lean women

57
Confidence Interval or P-value?

• Confidence interval!!!
• P-value will tell you whether or not there is a
  statistically significant difference
• confidence interval will give information about
  the size of the difference and the strength of
  the evidence

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Paired t-test

• Obvious pairing between observations
• two measurements on each subject (before-after
  study)
• case-control pairs
• Assumption - paired data are normally distributed
• Example - Systolic blood pressure (SBP) measured
  in 16 middle aged men before and after a standard
  exercise. Post-exercise SBP - Pre-exercise SBP
  calculated for each man

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Boxplot of differences
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Paired t-test

• Mean difference 6.6
• SE(Mean) 1.5
• t 6.6/1.5 4.4
• Compare with t(15)
• P lt 0.001
• Conclusion- mean systolic blood pressure is
  higher after exercise than before

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Paired t-test

• 95 confidence interval for the mean difference
• 6.6 ? 2.131.5 3.4 to 9.8

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Categorical Variables

• To investigate the relationship between two
  categorical variables form contingency table
• Hypothesis tests
• Chi-squared test (?2 test)
• Fishers exact test (small samples)
• McNemars test (paired data)

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Chi-squared test

• Used to test for associations between categorical
  variables (2 or more distinct outcomes)
• Example - a comparison between psychotherapy and
  usual care for major depression in primary care

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Patient Reported Recovery at 8 months
65
Patient Reported Recovery at 8 months

• Difference between means 30.8
• 95 confidence interval for difference 17.7 to
  43.8

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Larger tables

• Similar methods can be applied to larger tables
  to test the association between two categorical
  variables
• Example - Is there an association between housing
  tenure and time of delivery of baby
  (preterm/term).
• Null hypothesis There is no relationship
  between housing tenure and time of delivery

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Relationship between housing tenure and time of
delivery
68
Relationship between housing tenure and time of
delivery

• DF (5-1)??(2-1) 4
• P 0.03
• Thus we strong evidence of a relationship between
  housing tenure and time of delivery

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Notes

• Chi-squared test not valid if expected values are
  small (lt5)
• Combine rows or columns to obtain a smaller table
  with larger expected values
• Use Fishers exact test for small tables

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McNemars test

• Appropriate for use with paired or matched
  (case-control) data with a dichotomous outcome

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Example - McNemars test

• Skaane compared the use of mammography and
  ultrasound in the assessment of 327 (228 palpable
  and 99 non-palpable) consecutive malignant
  tumours confirmed at histology.
• Acta radiologica vol 40486-490 (1999)

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McNemars test - example
73
McNemars test - example

• 308/327 (94) were picked up by mammograpy
  compared with 278/327 (85) picked up by
  ultrasound
• Plt0.001
• Conclusion Mammography is significantly more
  sensitive in diagnosing tumours than ultrasound
  in a population of mixed malignant tumours

74
Hypothesis testing - summary
Adapted from Chinn S. Statistics for the European
Respiratory Journal.
75
Correlation and Regression

• Relationship between two continuous variables
• regression
• correlation

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Relationship between two continuous variables

• 3 main purposes for doing this
• to assess whether the two variables are
  associated (correlation)
• to enable the value of one variable to be
  predicted from any known value of the other
  variable (regression)
• to assess the amount of agreement between two
  variables (method comparison study)

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Example

• Women from a pre-defined geographical area were
  invited to have their haemoglobin (Hb) level and
  packed cell volume measured. They were also
  asked their age.

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Haemoglobin and packed cell volume
79
Example - relationships between variables

• Association between Hb and PCV? Hb affects PCV
  or PCV affects Hb?
• Use correlation to measure the strength of an
  association
• Association between Hb and age?age must affect
  Hb and not vice versa
• Use regression to predict Hb from age

80
Correlation

• Not interested in causation i.e. does a high
  PCV cause a high Hb level
• Interested in associationi.e. is a high PCV
  associated with a high Hb level?
• sample correlation coefficient
• summarises strength of relationship
• can be used to test the hypothesis that the
  population correlation coefficient is 0

81
Correlation Coefficient

• dimensionless, from -1 to 1
• measures the strength of a linear relationship
• ve - high value of one variable associated with
  high value of the other
• -ve - high value of one variable associated with
  low value of the other
• 1 exact linear relationship
• strictly called Pearson correlation coefficient

82
Example Data
r -0.4
r 1
r 0
r 0.7
83
When not to use the correlation coefficient

• If the relationship is non-linear
• with caution in the presence of outliers
• when the variables are measured over more than
  one distinct group (i.e. disease groups)
• when one of the variables is fixed in advance
• Assessing agreement

84
Correlation - example data
85
Is there an alternative?

• If the data are non-linear or there is an outlier
• use spearman rank correlation coefficient

86
Haemoglobin and Packed Cell Volume

• Without outlier
• Pearson0.67
• Spearman0.63
• With outlier
• Pearson0.34
• Spearman0.48

87
Regression

• Assume a change in x will cause a change in y
• predict y for a given value of x
• usually not logical to believe y causes x
• y is the dependent variable (vertical axis)
• x is the independent variable (horizontal axis)

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Example - Haemoglobin vs Age
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Regression

• Logical to assume that increasing age leads to
  increasing Hb
• Not logical to assume Hb affects age!
• Assume underlying true linear relationship
• Make an estimate of what that true linear
  relationship is

90
Estimating a regression line

• How do I identify the best straight line?
• least squares estimate
• straight line determined by slope and intercept
• y a b?x
• a and b are estimates of the true intercept and
  slope and are subject to sampling variation

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Regression line of haemoglobin on age
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Regression of haemoglobin on age

• Variable(s) Entered on Step Number 1.. AGE
  Age (Years)Multiple R .87959R
  Square .77367Adjusted R Square
  .76110Standard Error 1.17398
• Analysis of Variance DF
  Sum of Squares Mean SquareRegression
  1 84.80397
  84.80397Residual 18
  24.80803 1.37822F 61.53133
  Signif F .0000

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Regression of haemoglobin on age

• ---------------------- Variables in the Equation
  -------------Variable B SE B
  95 Confdnce Intrvl BAGE .134251
  .017115 .098295 .170208 (Constant)
  8.239786 .794261 6.571104 9.908467
• ----------- in ------------Variable T
  Sig TAGE 7.844 .0000(Constant)
  10.374 .0000

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What does this tell us?

• Mean Hb 8.2 0.13 ??AGE
• 95 CI for the slope goes from 0.098 to 0.170
• P lt 0.0001
• Significant relationship between Hb and age
• 77 of the variability in Hb can be accounted for
  by age

95
How can it be used?

• Predict mean Hb for a given age
• Eg. What is the mean Hb of a 50 year old?
• Mean Hb 8.2 0.13???50 14.7 g/dl
• 95 CI for the estimate from 14.4 to 15.5 g/dl

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How can it be used?

• To calculate reference ranges for the population
• E.g. What range would you expect 95 of 50 year
  olds to lie within? (reference range)
• Between 12.4 to 17.5 g/dl

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95 Confidence Interval for the Mean 95
prediction interval for individuals
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Definitions

• Predicted value
• the value predicted by the regression line
• an estimate of the mean value
• Residual
• Observed value - predicted value

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What assumptions have I made?

• The relationship is approximately linear
• The residuals have a normal distribution

100
Multiple Regression

• One outcome variable with multiple predictor
  variables
• Residuals assumed to be normally distributed
• Predictor variables can be continuous or
  categorical
• No assumptions made about distribution of
  continuous predictor variables

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Multiple Regression

• Example. Does the value of packed cell volume
  improve the prediction of hb?
• Model fitted
• Mean Hb 5.2 0.1?age(years) 0.1?packed
  cell volume()
• R2 83
• Knowledge of packed cell volume improves the
  prediction of haemoglobin

102
Summary

• Regression can be used to estimate the numerical
  relationship between an outcome variable and one
  or more predictor variables
• Correlation coefficient alone is of limited use