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Intro to Statistics Part2
Arier LeeUniversity of Auckland
• Standard error – the standard deviation of the sampling distribution of a statistic
• The standard deviation of the sample means is called the standard error of the mean and it measures how precisely the population mean is estimated by the sample mean
• The standard error is a measure of the precision of the estimated mean whereas the standard deviation summarises the variability or the spread of the observations
• Standard error <= standard deviation • The larger the sample size the smaller the standard error
Standard error
• A 95% confidence interval for a mean is calculated by
(mean-1.96*SE, mean+1.96*SE)• An example: In a sample of 2000 pregnant
women, serum cholesterol was measured and it was found that the sample mean is 5.62 and SE=0.15. 95% confidence interval:
(5.33, 5.91)
Confidence intervals
• 95% CI does not mean that there is a 95% chance that the true mean lies between 5.33 and 5.91
• If we repeat the study over and over again, calculating a 95% confidence interval each time, about 95 of 100 such intervals would include the true mean
• Whether the one that we have obtained from our study is one of them we will never know – but we have some confidence
• It is a measure of precision of our estimate• Bigger confidence interval -> less precision
Confidence intervals
• Exploratory data analysis• Presentation of results• Examples: Bar charts, Line graphs, Scatter plots,
Box plots, Kaplan Meier Plots etc.• Graphs can only be as good as the data they
display• No amount of creativity can produce a good
graph from dubious data
Graphical presentation of the data
Bar chart
2005 maternity report
Line graph
Box plot
median
Q1
Q31.5 x (Q3-Q1)
Smallest obs marks end of whisker
Obs beyond end of whisker
Data to chart ratioMental health score by treatment groups
Good Bad
Inadequate chart type
0-14 years 15-24 years 25-64 years 65+ years0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Māori
Pacific
Asian
Other Ethnicity (reference)
Incid
ence
rate
ratio
Effect of ethnicity on road traffic injury deaths and hospitalisations, 2000-8, Auckland region, by age group, adjusted for gender and deprivation (using National Minimum Data Set and Mortality Collection data)
• Points with error bars• Log scale
Graphs of risk or rate ratio should be presented with
Odds ratio presented with logarithmic scale
Outcome: Blindness
New
Zea
land
Eur
opea
n
Māo
ri
Paci
fic P
eopl
es
Asia
n
Mid
dle
East
ern/
Latin
Am
eric
an/A
fric
an
Oth
er(N=2909)
(N=571)(N=483)
(N=548)(N=91)
(N=14)
0
10
20
30
40
50
60
70
80
90
100
Seldom or never
weekly
Daily
Perc
ent
(%)
Unnecessary 3D effects
How often do you read to your child
Inadequate labelling
ApplePearBanana
• Use appropriate graph types for the appropriate purpose, e.g. line chart for trend
• All axes, tick marks, title, should be labelled• Appropriate scale used• Adequate data to chart ratio• Avoid unnecessary complexity such as• Irrelevant decoration• Too much colours• 3D effects
• Keep it simple!
Graphical presentation of the data
Research process
Research question
Primary and secondary endpoints
Study design
Sampling and/or randomisation scheme
Power and sample size calculation
Pre-define analyses methods
Analyse data
Interpret results
Disseminate
• One of the statistical, economical and ethical issues of the design of medical studies• Statistical: Ensure the study is large enough to
detect an effect if it exists• Economical: Ensure not enlist more patients than
are needed• Ethical: unethical to engage more people in a trial
than are needed• Larger samples -> more precise estimates• How large?
Sample size and power of a study
• The power of a test is the probability of detecting a true difference
• The size of the sample needed depends on• required power• detectable difference• variability in the population• level of significance (probability of falsely reject the
NULL)• statistical test being used
• Need information to calculated a meaningful sample size – literature search
Sample size and power of a study
• A double blind randomised controlled study on treatment for chronic hypertension during pregnancy
• Comparing two treatments:• Standard treatment• New treatment
Sample size and power of a study- an example
• Based on current evidence, assume– Detectable difference: 10mmHg– Standard deviation: 15 mmHg– 90% power– 5% significance level– Two-sided test– 1:1 ratio
• Using PS (a power and sample size calculation software) – 48 subjects per group
• After considering drop-out rate, say 10%, round to, say, 60 subjects per group
Sample size and power of a study- an example
Sample size and power of a studyChronic hypertension during pregnancy example
• To detect a difference of 10mmHg
• SD varies from 5 to 30mmHg
Sample size calculation is an evidence based best guess• Relies on assumptions• Not a precise number• No guarantee of significant effect at the end of a
study
Sample size and power of a study
Any Questions?