Session 6: Basic Statistics Part 1
(and how not to be frightened by the word)
Prof Neville Yeomans
Director of Research, Austin LifeSciences
So now we’ve got some
results? How can we
make sense out of them?
What I will cover
(over 2 sessions, August 28 and September 25)
• Sampling populations
• Describing the data in the samples
• How accurately do those data reflect the ‘real’
population the samples were taken from?
• We’ve compared two groups. Are they really from
different populations, or are they samples from the
same population and the measured differences are
just due to chance?
What I will cover (contd.)
• Tests to answer the question ‘Are the differences
likely to be just due to chance?’
– Data consisting of values (e.g. hemoglobin
concentration)(‘continuous variables’)
– Data consisting of whole numbers – frequencies,
proportions (percentages)
– Tests for just two groups; tests for multiple groups
• Tests that examine relationships between two or
more variables (correlation, regression analysis, lifetable)
What I will cover (contd.)
• How many subjects should I study to find the answer
to my question? (Power calculations)
• Statistical packages and other resources
We’ve got some numbers. How are we
going to describe them to others?
Suppose we’ve measured heights of a number of
females (‘a sample’) picked off the street in Heidelberg
Subject
#
Height
(cm)
Subject
#
Height
(cm)
Subject
#
Height
(cm)
1
179
9
156
17
167
2
162
10
157
18
173
3
150
11
161
19
159
4
155
12
164
20
170
5
168
13
165
21
168
6
175
14
159
22
162
7
159
15
161
23
171
8
152
16
163
24
164
How could
H ewe
ig h tsmore
o f (s aconcisely
m p le o f) H edescribe
id e lb e rg wthese
o m e n data –
using just one or two numbers that would give us
useful information about the sample as a whole?
185
180
H e ig h t (c m )
175
170
165
160
155
150
1. A measure of ‘central tendency’
P e rso n n u m
b e r values are spread
2. A measure of how widely
the
145
0
2
4
6
8
10
12
14
16
18
20
22
24
26
F re q u e n cy o f w o m e n w ith e a ch h e ig h t in sa m p le
The median (middle value) = 162.5
3 .5
The range (150-179)
a poor measure for describing the
whole population because it depends
on sample size – range is likely to be
wider with larger samples
3 .0
F re q u e n cy
2 .5
2 .0
Interquartile range
(25th percentile to 75th
percentile of values: 159-168)
1 .5
what we should always
use with the median –
it’s largely independent
of sample size
1 .0
0 .5
145
150
155
160
165
H e ig h t
170
175
180
185
F re q u e n c y d is trib u tio n o f h e ig h t o f H e id e lb e rg w o m e n
(amalgamated into 3cm ranges: e.g.141-143, 144-146 etc.)
the Mean (average)(Ʃx/N) = 163.3 cm
6
the Standard Deviation*
± 7.2 cm
5
- doesn’t vary much with sample size
(except very small samples)
- approx. 67% of values will lie within
± 1 SD either side of mean**
- approx 95% of values will lie within
± 2 SD either side of mean**
F re q u e n cy
4
3
2
1
0
130
140
150
160
170
180
190
H e ig h t
*In Excel, enter formula ‘=STDEV(range of cells)’
** Provided the population
is ‘normally distributed’
200
The ‘Normal distribution’
Mean = Median in a ‘perfect’
normal distribution
Standard deviations away from mean
We measured the mean height of our
sample of 25 women ... (it was 163.3 cm)
• But what is the average height of the whole
population – of ALL Heidelberg women?
• We didn’t have time or resources to track them
all down – that’s why we just took what we
hoped was a representative sample.
• What I’m asking is: how good an estimate of the
true population mean is our sample mean?
• This is where the Standard Error of the Mean* (or
just Standard Error, SE) comes in.
*It’s sometimes called the Standard ESTIMATE of the Error of the mean
The Standard Error (contd.)
• The mean height of our sample of 25 women was 163.3 cm
• We calculated the Standard Deviation (SD) of the sample to be
7.3 cm (that value, on either side of the mean, that should
contain about 2/3 of those measured)
• Standard Error of the mean = SD/√N , i.e. 7.3/ √25 = 7.3/5 =
1.46
• So now we can express our results for the height of our
sample as 163.3 ± 1.5 (Mean ± SEM) ...... But what does this
really tell us?
• The actual true mean height of the whole population of
women has a 67% likelihood of lying within 1.5 cm (i.e. 1 SEM)
either side of the mean we found in our sample; and a 95%
likelihood of lying within 3 cm (i.e. 2 x SEM) either side of that
sample mean.
(It’s actually 1.96xSEM for a reasonably large sample - e.g.
roughly 30 – and wider for small samples, but let’s keep it simple).
The concept of the standard error of the
mean (SEM) – e.g. serum sodium values
True population mean
142 mmol/L (SD=4.0)
Sample mean
=142.8 mmol/L
(SD of sample = 3.4)
1 x SEM = (i.e. 3.4/√10)
= 1.1 mmol/L
Random sample of 10
normal individuals
130
2 x SEM (~’95% confidence interval’)
= 2.2 mmol/L
That means: ‘There is a 95% chance (19 chances out of 20) that
the actual population mean, estimated from our random sample
lies between 140.6 and 145.0 mmol/L)’
134
138
142
146
150
154 mmol/L
Why does Standard Error depend
on population SD and sample size?
SE = SD/√N
A: Narrow population
spread (i.e. small SD)
B: Wide population
spread (i.e. large SD)
Increasing N decreases SE of mean.
i.e. increases accuracy of our estimate of the population mean
based on results of our sample
Testing significance of differences
S a m p le s o f H e id e lb e rg m e n a n d w o m e n
7
6
Women:
Mean = 163.3
SE = 1.46
95% confidence intervals
F re q u e n cy
5
4
3
Men:
Mean = 176.5
SE = 1.43
2
On a quick, rough,1 check we can see that:
(a) the 95% confidence interval for our estimate of the height of women is
0
160.3-166.4 cm (approximately
mean
± 2SE).
130
140
150
160
170
180
190
200
(b) our estimate of the mean height of the men sampled is quite a lot outside
H e ig h t (cm )
the 95% confidence interval (range)for the women, so it looks improbable that
they are from the same population
Testing significance of differences ....
How likely is it that the two random samples
came from the same population?
Student’s t-test
Composite frequency
distribution, created by pooling
data from both samples
Women
Men
Mean: 170.7
SD: 10.0 cm
How likely is it that these two samples (the pink and the blue) were taken from
the SAME population? [this is called the NULL HYPOTHESIS]
Tested for statistical significance of difference: p<0.001
i.e. there is less than 1 chance in 1000 that these two samples
-2
0
1
2
3
came-3from the
same -1population
Standard deviation either side of mean
In fact, though, running a Student’s
t-test on the two samples of height in the Heidelberg
men and women slide, gives this error message:
Assumptions to be met before testing
significance of differences with PARAMETRIC
TESTS* – i.e. tests that use the mathematics of
the normal curve distribution
• The combined data should approximate a normal
curve distribution
– in this instance the male data were skewed (not
evenly distributed around the mean) and spread a
bit too far out into the tails of the frequencydistribution curve
• The variances (=SD2) of the groups should not differ
significantly from each other
*Student’s t-test, Paired t-test, Analysis of Variance
An example of data where groups have
different variance (spread), and one group is
skewed
obstruction - controls vs smokers
The equal varianceAirway
test failed
(P<0.05), and the normality test almost
100
failed (P=0.08)
– so we should not use a parametric test such as t-test
Means
90
Means
FEV1/FVC
80
Lower limit of
‘normal’ range
70
Medians
60
50
40
30
Controls
Smokers
So what do we do if we can’t use a
parametric test to check for
significances of differences?
• Use a non-parametric test
• These tests, instead of using the actual numerical
values of the data, put the data from each group into
ascending order and assign a rank number for their
place in the combined groups.
• The maths of the test is then done on these ranks
• Examples: Rank sum test, Wilcoxon Rank Test, Mann
Whitney rank test, etc.
– (the P value for our slide of heights of Heidelberg men and
women was calculated using Wilcoxon test)
Example of how a rank-sum (nonparametric) test is constructed manually*
Group 1
data
12
13
13
15
19
27
Sum of
ranks:
Rank when
groups
combined
1
2.5
2.5
4
6.5
9
Group 2
data
16
19
26
40
78
101
Rank when
groups
combined
25.5
5
6.5
8
10
11
12
46.5
Mann-Whitney test: P = 0.026
*In reality, these days you’ll just feed the raw data into a program to
do it for you
Tests to examine significance of
differences between 3 or more groups
[strictly this should read ... ‘tests to decide how
likely are data from 3 or more samples to come from
the same population’]
• Parametric tests (tests based on the
mathematics of the ‘normal curve’)
– Analysis of Variance (1-way, 2-way, factorial, etc.)
• Non-parametric tests (rank-sum tests)
– Kruskal-Wallis test
P a ra lle l g ro u p tria l o f p la c e b o v e rs u s N o rm o p re s s tre a tm e n t
fo r h yp e rte n s io n
140
M e a n 2 4 h o u r b lo o d p re ssu re
M e a n + /- S E
120
100
80
60
40
20
0
0 mg
0 .5 m g
N o rm o p re s s d o s e
2 .0 m g
P a ra lle l g ro u p tria l o f p la c e b o v e rs u s N o rm o p re s s tre a tm e n t
fo r h yp e rte n s io n
140
M e a n 2 4 h o u r b lo o d p re ssu re
M e a n + /- S E
120
100
One variable (dose), compared
across 3 groups .... So this gets
tested with one-way ANOVA
80
60
40
20
0
0 mg
0 .5 m g
N o rm o p re s s d o s e
2 .0 m g
P a ra lle l g ro u p tria l o f p la c e b o v e rs u s N o rm o p re s s tre a tm e n t
fo r h yp e rte n s io n
140
M e a n 2 4 h o u r b lo o d p re ssu re
M e a n + /- S E
120
100
80
60
40
20
0
0 mg
0 .5 m g
2 .0 m g
N o rm o p re s s d o s e
This tells us that it is very unlikely the
three groups belong to the same
population.
.
But which differ from which?
One Way Analysis of Variance
Normality Test:
Passed
Equal Variance Test: Passed
Group Name
Control
Normopress 0.5 mg
Normopress 2.0 mg
Source of Variation
Between Groups
Residual
Total
Thursday, June 28, 2012, 3:39:25 PM
(P = 0.786)
(P = 0.694)
N Missing
12
0
12
0
12
0
DF
2
33
35
Mean
Std Dev
126.500
7.243
125.083
5.551
114.000
5.625
SS
MS
1124.389 562.194
1263.917 38.301
2388.306
F
14.679
SEM
2.091
1.602
1.624
P
<0.001
The differences in the mean values among the treatment groups are greater than would be expected by
chance; there is a statistically significant difference (P = <0.001).
All Pairwise Multiple Comparison Procedures (Holm-Sidak method):
Overall significance level = 0.05
Comparisons for factor:
Comparison
Diff of Means
t
Unadjusted P
Control vs. Normopress 2.0 mg
12.500
Normopress 0.5 mg vs. Normopress 2.0 mg 11.083
Control vs. Normopress 0.5 mg
1.417
4.947
4.387
0.561
<0.001
<0.001
0.579
Critical Significant?
Level
0.017
Yes
0.025
Yes
0.050
No
Before and after data – paired tests
• Create a paired analysis of length of hair after
going to hairdresser.
• Hypothesis: cutting hair makes it shorter
Two independent groups
Difference between means tested for significance with
Student’s t test
20
P=0.58
Hair length (cm)
15
10
9.7±1.9*
8.2±1.7
5
*mean ± SE
0
1
2
Group
Our actual data
Difference between means tested for significance
with paired Student’s t test
The variation within each individual is much less than between individuals
20
P=0.025
Hair length (cm)
15
10
5
The paired t-test examines the mean and standard error of the changes
In each individual, and tests how likely are the changes due to chance
0
1
Before
2
After
Group
So far we have been dealing with
‘CONTINUOUS VARIABLES’
– numbers such as heights, laboratory values,
velocities, temperatures etc. that could have any
value (e.g. many decimal points) if we could measure
accurately enough.
Now we’ll look at ...
‘DISCONTINUOUS VARIABLES’
– whole numbers, most often as proportions
or percentages.
Rates and proportions
• In 1969, a home for retired pirates has 93
inmates, 42 of whom have only one leg.
• In 2004, a subsequent survey finds there are
now 62 inmates, 6 of whom have only one leg.
Has there been a ‘real’ change (i.e. a change
unlikely to be due to chance) in the
proportion of one-legged pirates in the home
between the two surveys?
Year
Pirates with 1 leg
Pirates with 2 legs
(%)
Expected (%)
1969
42 (45.2)
29
51 (54.8)
93
2004
6 (9.7)
19
56 (10.3)
62
Totals
48
107
Total pirates
155
Chi-square= 20.282 with 1 degrees of
freedom. (P = <0.001)
i.e. The likelihood that the difference
in proportions of 1-legged inmates
between 1969 and 2004 is due to
chance ... is less than 1:1000
One trap with chi-square tests and small
numbers ....
Penicillin treatment for pneumonia
Treatme
TreatmentNo.No.
dead
deadNo. No. Totals
nt
surviving
surviving
Placebo
Placebo
9 9
5
5
14
Penicillin
Penicillin
1 1
8
8
9
Totals
10
13
23
Fisher’s exact test:
P = 10! x 13! x 14! x 9!
9! x 5! x 1! x 8! x 23!
= 0.029
Correlation
• Fairly straightforward concept of how likely are two
variables to be related to each other
• Examples:
– Do children’s heights vary with their age, and if so is the
relation direct (i.e. get bigger as get older) or converse (get
smaller as get older)?
– Does respiratory rate increase as pulse rate increases
during exertion?
• The correlation coefficient, R, tells us how closely the
two variables ‘travel together’
• P value is calculated to tell us how likely the
relationship is to be ‘only’ by chance
Examples of regression
(correlation) data
22
18
20
R = 0.965
P<0.0001
14
18
16
12
Y Data
M W g e n e ra te d
16
10
14
12
10
R = - 0.37
P = 0.30
8
8
6
6
4
4
0
2
4
6
8
H o u rs o f s u n lig h t
10
12
0
2
4
6
X Data
8
10
12
Some other common statistical
analyses
• Life-table analyses
– Observing and comparing events developing over
time; allows us to compensate for dropouts at
varying times during the study
• Multiple linear and multivariate regression
analyses
– Looking for relationships between multiple
variables
Life table analyses
Advanced lung cancer. Trial compared motesanib + 2 conventional
chemo drugs ... with placebo plus the two other drugs
Scagliotti et al. J Clin Oncol 2012; 30: 2829
Multiple regression analysis
• Examines the possible effect of more than one
variable on the thing we are measuring (the
‘dependent variable’)
Perret JL et al. The Interplay between the Effects of Lifetime
Asthma, Smoking, and Atopy on Fixed Airflow Obstruction in
Middle Age. Am J Respir Crit Med 2013; 187: 42-8.
...from Institute of Breathing and Sleep (Austin),
University of Melbourne, Monash University, Alfred Hospital,
And others
Perret et al. 2013
Sample Size Calculations
How many patients, subjects, mice
etc. do we need to study to reliably*
find the answer to our research
question?
*We can never be certain to do this, but should aim to be considerably
more likely than not to find out the truth about the question
Sample size calculations (1)
First we need to grapple with two types of ‘error’ in
interpreting differences between means and/or
medians of groups:
• Type 1 (or α) error: ... that we think the difference is
‘real’ (data are from 2 or more different populations)
when it is not
– This is what we’ve dealt with so far, and the Pvalues assess how likely the differences are due to
chance
• Type 2 (or β) error: ... that our experiment, and the
stats test we’ll apply to the results, will FAIL to show
a significant difference when there REALLY IS ONE
Sample size calculations (2)
• If we end up with a Type 2 () error, it will be
because our sample size(s) was too small to
persuade us that the actual difference
between means was unlikely due to chance
(i.e. P<0.05)
• The smaller the real difference between
population means, the larger the sample size
needs to be to detect it as being statistically
significant
Sample size calculations (3)
How do we go about it?
Most of the good statistical packages have a function for
calculating sample sizes
1. Decide what statistical test will be appropriate to apply
to primary endpoint when study completes
2. Estimate the likely size of difference between groups, if
the hypothesis is correct
3. Decide how confident you want to be that the
difference(s) you observe is unlikely due to chance
4. Decide how much you want to risk missing a true
difference (i.e. what power you want the study to have)
Note: We really should have done a sample size calculation before we started our
experiments, but for this course we needed to deal with the basics of stats tests first
Sample size calculations
A worked example (i)
•
We want to see whether drug X will reduce the
incidence of peptic ulcer in patients taking aspirin for
6 months
1. Decide what statistical test: chi square, to
compare differences in frequencies in 2 groups
Sample size calculations
A worked example (i)
•
•
•
We want to see whether drug X will reduce the
incidence of peptic ulcer in patients taking aspirin for
6 months
We expect a 10% incidence of ulcers in the controls
We hypothesize that a 50% reduction (i.e. 5%) in
those treated with X would be clinically worthwhile
2. We’ve now decided the size of the difference
between groups we are interested to look for
Sample size calculations (4)
A worked example (i)
•
•
•
•
We want to see whether drug X will reduce the
incidence of peptic ulcer in patients taking aspirin for
6 months
We expect a 10% incidence of ulcers in the controls
We hypothesize that a 50% reduction (i.e. 5%) in
those treated with X would be clinically worthwhile
We decide to be happy with a likelihood of only 1:20
that difference observed is due to chance
3. That is to say, we want to set P0.05 as the level
of α (alpha) risk (the risk of concluding the
difference is real when it’s actually due to chance)
Sample size calculations (4)
A worked example (i)
•
•
•
•
•
We want to see whether drug X will reduce the
incidence of peptic ulcer in patients taking aspirin for
6 months
We expect a 10% incidence of ulcers in the controls
4.We
That
is, set Power
≥80%(i.e.
(1-β)
to in
hypothesize
thatofa the
50%study
reduction
5%)
detect such a difference (if it exists)
those treated with X would be clinically worthwhile
We decide to be happy with a likelihood of only 1:20
that difference observed is due to chance
We would like to have at least an 80% chance of
finding that 50% reduction (20% of missing it, i.e. of
β risk)
Sample size calculations
A worked example (i)
Summary of sample size calculation setting:
• Estimated ulcer incidence in controls = 10%
• Estimated incidence in group receiving drug X
= 5%
• For P(α) 0.05, and Power (1-β) ≥80%
• Data tested by chi square
......................................................................
Calculated required sample size = 449 in each
group
Sample size calculations
A worked example (ii)
•
•
•
•
•
We hypothesize that removing the spleen in rats will
result in an increase in haemoglobin (Hb) from the
normal mean of 14.0 g/L to 15.0 g/L
We already know that the SD (Standard deviation) of Hb
values in normal rats is 1.2 g/L (if we don’t know we’ll
have to guess!)
Testing will be with Student’s t test
We’ll set α (likelihood observed difference due to
chance) at 0.05
We want a power (1-β) of at least 80% to minimize risk
of missing such a difference if its real*
*More correctly, we should say if the samples really are from different populations
Sample size calculations
A worked example (ii)
Summary of sample size calculation setting:
• Control mean = 14.0 g/L; Operated mean =
15.0 g/L
• Estimated SD in both groups = 1.2 g/L
• For P(α) 0.05, and Power (1-β) ≥80%
• Data tested by Student t
......................................................................
Calculated required sample size = 24 in each
group
Summary of the most common statistical
tests in biomedicine (1. Parametric tests)
Test
Purpose
Comments
Student t-test
Compare 2 groups of
‘continuous’ data*
Only use if data are ‘normally
distributed’ and variances of
groups similar
Paired Student t-test
Compare before-after
data on the same
individuals
The differences (between beforeafter) need to be ‘normally
distributed’
More powerful than ‘unpaired’ ttest because less variability within
individuals than between them
1- way analysis of
variance
Compare 3 or more
groups of continuous
data
Same requirements as for Student
t-test
2-way analysis of
variance
Compare 3 or more
groups , stratified for at
least 2 variables
As above
*For measured values, not numbers of events (frequencies)
Summary of the most common statistical
tests in biomedicine (2. non-parametric)
Test
Purpose
Comments
Rank-sum or MannWhitney test
Compare 2 groups of
‘continuous’ data, using
their ranks rather than
actual values
Use if t-test invalid because data
not ‘normally distributed’ and/or
variances of groups significantly
different
Signed rank test
Rank test to use instead
of paired t-test
Use instead of paired t-test if the
differences (between before and
after) are not ‘normally
distributed’
Non-parametric
analysis of variance
Compare 3 or more
groups of continuous
data
As above (it’s the generalised form
of Mann-Whitney test when there
are >2 groups)
Some tools for statistical analyses
• Excel spreadsheets – e.g. If column A contains
data in the 8 cells, A3 through A10
– Mean :
– SD:
– SEM:
=average(a3:a10)
=stdev(a3:a10)
=(stdev(a3:a10))/sqrt(8)
• Common statistical packages for significance
testing
– Sigmaplot
– SPPS (licence can be downloaded from Unimelb)
– STATA
Other resources
• Armitage P, Berry G, Matthews JNS. Statistical methods in
medical research. 4th edn. Oxford: Blackwell Science, 2002.
• Dawson B, Trapp R. Basic and clinical biostatistics. 4th edn.
New York: McGraw Hill 2004. (Electronic book in Unimelb
electronic collection)
• Rumsey DJ. Statistics for Dummies. 2nd edn. Oxford: Wiley &
sons 2011.