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Introduction
• Statistics are increasingly prevalent in medical
practice, and for those doing research,
statistical issues are fundamental. It is
extremely important therefore, to understand
basic statistical ideas relating to research design
and data analysis, and to be familiar with the
most commonly used methods of analysis.
• Although data analysis is certainly an important
part of the statistical process, there is an
equally vital role to be played in the design of
the research project. Without a properly
designed study, the subsequent analysis may be
unsafe, and/or a complete waste of time and
resources.
•
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•
•
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•
•
•
•
•
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•
Types of data
Descriptive statistics
Data distributions
Comparative statistics
Non-parametric tests
Paired data
Comparison of several means
Comparing proportions
Exploring the relationship between 2 variables
Correlation
Linear regression
Survival analysis
Proportion of total
.15
.1
900
.05
0
0 100
400
1000
Platelet count
1500
Types of Data
• Categorical
– binary or dichotomous e.g. diabetic/non-diabetic,
smoker/non-smoker
– nominal e.g. AB/B/AB/O, short-sighted/longsighted/normal
– ordered categorical (ordinal) e.g. stage 1/2/3/4,
mild/moderate/severe
• Discrete numerical - e.g. number of children 0/1/2/3/4/5+
• Continuous - e.g. Blood pressure, age
• Other types of data
–
–
–
–
–
–
ranks, e.g. preference between treatments
percentages, e.g. % oxygen uptake
rates or ratios, e.g. numbers of infant deaths/1000
scores, e.g. Apgar score for evaluating new-born babies
visual analogue scales, e.g. perception of pain
survival data – two components, outcome and time to
outcome
Descriptive Statistics
• For continuous variables there are a number of useful
descriptive statistics
– Mean - equal to the sum of the observations divided by the
number of observations, also known as the arithmetic mean
– Median - the value that comes half-way when the data are
ranked in order
– Mode - the most common value observed
– Standard Deviation - is a measure of the average deviation
(or distance) of the observations from the mean
– Standard Error of the mean - is measure of the uncertainty
of a single sample mean as an estimate of the population
mean
Data Distributions
• Frequency distribution
– If there are more than about 20 observations, a
useful first step in summarizing quantitative data is
to form a frequency distribution. This is a table
showing the number of observations at different
values or within certain ranges. If this is then
plotted as a bar diagram a frequency distribution is
obtained.
20
Std. Dev = 11.43
Mean = 34.3
0
N = 1712.00
0
6.
-6
.0 .0
65 - 62
.0 .0
61 - 58
.0 .0
57 - 54
.0 .0
53 - 50
.0 .0
49 - 46
.0 .0
45 - 42
.0 .0
41 - 38
.0 .0
37 - 34
.0 .0
33 - 30
.0 .0
29 - 26
.0 .0
25 - 22
.0 .0
21 - 18
.0
17
10
Frequency
Histogram of patient ages for HD
80
70
60
50
40
30
PAGE
The Normal Distribution
• In practice it is found that a reasonable
description of many variables is provided by
the normal distribution (Gaussian distribution).
The curve of the normal distribution is
symmetrical about the mean and bell-shaped.
The bell is tall and narrow for small standard
deviations, and short and wide for large ones.
Frequency
paraprotein in myeloma
20
18
16
14
12
10
8
6
4
Std. Dev = 18.81
2
Mean = 33.0
0
N = 103.00
0.0
10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0
5.0
15.0 25.0 35.0 45.0 55.0 65.0 75.0 85.0
PARAP
20
18
16
14
12
10
8
6
4
Std. Dev = 18.81
2
Mean = 33.0
0
N = 103.00
0.0
10.0
5.0
PARAP
20.0
15.0
30.0
25.0
40.0
35.0
50.0
45.0
60.0
55.0
70.0
65.0
80.0
75.0
90.0
85.0
Duration of disease pre ASCT for HD
700
600
500
400
300
Frequency
200
Std. Dev = 3.90
100
Mean = 3.2
N = 1676.00
0
.0
32
.0
30
.0
28
.0
26
.0
24
.0
22
.0
20
.0
18
.0
16
.0
14
.0
12 0
.
10
0
8.
0
6.
0
4.
0
2.
0
0.
Duration of disease (y)
Descriptives
DOD
Mean
95% Confidence
Interval for Mean
5% Trimmed Mean
Median
Variance
Std. Deviation
Minimum
Maximum
Range
Interquartile Range
Skewness
Kurtosis
Lower Bound
Upper Bound
Statistic
3.1813
2.9946
Std. Error
9.515E-02
3.3679
2.5920
1.8000
15.174
3.8954
.10
33.40
33.30
2.3000
3.115
12.507
.060
.119
Comparative statistics
• When there are two or more sets of observations
from a study there are two types of design that must
be distinguished: independent or paired. The design
will determine the method of statistical analysis
• If the observations are from different groups of
individuals, e.g. ages of males and females, or
spectacle use in diabetics/non-diabetics, then the
data is independent. The sample size may vary from
group to group
• If each set of observations is made on the
same group of individuals, e.g. WBC count preand post- treatment, then the data is said to be
paired. This indicates that the observations are
on the same individuals rather than from
independent samples, and so we have the same
number of observations in each set of data
Independent data
• With independent continuous data, we are
interested in the mean difference between the
groups, but the variability between subjects
becomes important. This is because the two
sample t test (the most common test used), is
based on the assumption that each set of
observations is sampled from a population with
a Normal Distribution, and that the variances
of the two populations are the same.
Non-parametric test
• If the continuous data is not normally distributed, or
the standard deviations are very different, a nonparametric alternative to the t test known as the
Mann-Whitney test can be utilised (another
derivation of the same test is due to Wilcoxon)
30
93
25
20
15
10
BET2MG
109
107
94
52
7
111
5
0
N=
4
31
67
19
1.00
2.00
3.00
4.00
MMSTAGE
T-test
Group Statistics
BET2MG
MMSTAGE
3.00
4.00
N
67
19
Mean
3.6006
7.5179
Std. Error
Mean
.17805
1.18119
Std. Deviation
1.45738
5.14869
Independent Samples Test
Levene's Test for
Equality of Variances
F
BET2MG
Equal variances
ass umed
Equal variances
not as sumed
11.739
Sig.
.001
t-tes t for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower
Upper
-5.559
84
.000
-3.9173
.70463
-5.31852
-2.51607
-3.279
18.825
.004
-3.9173
1.19453
-6.41906
-1.41553
Mann-Whitney Test
Ranks
BET2MG
MMSTAGE
3.00
4.00
Total
N
67
19
86
Mean Rank
36.78
67.18
Sum of Ranks
2464.50
1276.50
Test Statisticsa
Mann-Whitney U
Wilcoxon W
Z
Asymp. Sig. (2-tailed)
BET2MG
186.500
2464.500
-4.685
<0.0001
a. Grouping Variable: MMSTAGE
Group Statistics
MMSTAGE
1.00
2.00
BET2MG
N
Mean
2.1050
2.7852
4
31
Std. Deviation
.40673
.96692
Std. Error
Mean
.20337
.17366
Independent Samples Test
Levene's Test for
Equality of Variances
F
BET2MG
Equal variances
ass umed
Equal variances
not as sumed
.964
Sig.
t-tes t for Equality of Means
t
.333
df
Sig. (2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower
Upper
-1.377
33
.178
-.6802
.49411
-1.68544
.32512
-2.543
8.518
.033
-.6802
.26743
-1.29039
-.06993
Test Statisticsb
Mann-Whitney U
Wilcoxon W
Z
Asymp. Sig. (2-tailed)
Exact Sig. [2*(1-tailed
Sig.)]
BET2MG
27.000
37.000
-1.816
.069
.073
a
a. Not corrected for ties .
b. Grouping Variable: MMSTAGE
60
3
577
50
444
43
7
40
416
83
327
30
NEUTS
20
10
0
N=
TBIDOSE
141
122
12
14.4
Neutrophil engraftment following allogeneic SCT for CML
Valid
NEUTS
TBIDOSE
12
14.4
N
141
122
Percent
88.1%
95.3%
Cases
Missing
N
Percent
19
11.9%
6
4.7%
Total
N
160
128
Percent
100.0%
100.0%
Descriptives
NEUTS
TBIDOSE
12
14.4
Mean
95% Confidence
Interval for Mean
5% Trimmed Mean
Median
Variance
Std. Deviation
Minimum
Maximum
Range
Interquartile Range
Skewness
Kurtosis
Mean
95% Confidence
Interval for Mean
5% Trimmed Mean
Median
Variance
Std. Deviation
Minimum
Maximum
Range
Interquartile Range
Skewness
Kurtosis
Lower Bound
Upper Bound
Lower Bound
Upper Bound
Statistic
22.9787
21.8289
Std. Error
.5816
24.1286
22.5816
22.0000
47.692
6.9060
11.00
56.00
45.00
9.0000
1.162
3.184
26.6148
25.5172
.204
.406
.5544
27.7123
26.1184
26.0000
37.495
6.1233
15.00
53.00
38.00
7.2500
1.453
4.157
.219
.435
50
50
40
40
30
30
20
20
10
10
0
0
10.0
NEUTS
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
55.0
15.0
NEUTS
20.0
25.0
30.0
35.0
40.0
45.0
50.0
55.0
200
3
327
87
416
371
444
294
66
43
190
462
561
100
135
302
62
PLATES
287
9
434
108
263
0
N=
TBIDOSE
128
98
12
14.4
Platelet engraftment following allogeneic SCT for CML
PLATES
TBIDOSE
12
14.4
Valid
N
Percent
128
80.0%
98
76.6%
Cas es
Mis sing
N
Percent
32
20.0%
30
23.4%
Total
N
Percent
160
100.0%
128
100.0%
Descriptives
PLATES
TBIDOSE
12
14.4
Mean
95% Confidence
Interval for Mean
5% Trimmed Mean
Median
Variance
Std. Deviation
Minimum
Maximum
Range
Interquartile Range
Skewness
Kurtosis
Mean
95% Confidence
Interval for Mean
5% Trimmed Mean
Median
Variance
Std. Deviation
Minimum
Maximum
Range
Interquartile Range
Skewness
Kurtosis
Lower Bound
Upper Bound
Lower Bound
Upper Bound
Statistic
32.7891
29.4857
Std. Error
1.6694
36.0924
30.5556
29.5000
356.703
18.8866
14.00
186.00
172.00
11.7500
5.244
37.479
42.8776
35.0417
.214
.425
3.9481
50.7134
37.1973
27.0000
1527.572
39.0842
14.00
185.00
171.00
18.0000
2.368
4.780
.244
.483
60
40
50
30
40
30
20
20
10
10
0
5.0
25.0
15.0
PLATES
45.0
35.0
65.0
55.0
85.0
75.0
105.0
95.0
125.0
115.0
145.0
135.0
165.0
155.0
185.0
175.0
195.0
0
10.0
30.0
20.0
PLATES
50.0
40.0
70.0
60.0
90.0
80.0
110.0
100.0
130.0
120.0
150.0
140.0
170.0
160.0
190.0
180.0
Group Statistics
NEUTS
PLATES
TBIDOSE
12
14.4
12
14.4
N
Mean
22.9787
26.6148
32.7891
42.8776
141
122
128
98
Std. Deviation
6.9060
6.1233
18.8866
39.0842
Std. Error
Mean
.5816
.5544
1.6694
3.9481
Independent Samples Test
Levene's Test for
Equality of Variances
F
NEUTS
PLATES
Equal variances
ass umed
Equal variances
not as sumed
Equal variances
ass umed
Equal variances
not as sumed
2.291
28.139
Sig.
.131
.000
t-tes t for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower
Upper
-4.486
261
.000
-3.6360
.8105
-5.2320
-2.0401
-4.525
260.837
.000
-3.6360
.8035
-5.2182
-2.0539
-2.557
224
.011
-10.0885
3.9448
-17.8622
-2.3148
-2.354
131.572
.020
-10.0885
4.2865
-18.5679
-1.6091
Test Statistics
Mann-Whitney U
Wilcoxon W
Z
P-value (2-tailed)
PLATES
6172.500
11023.500
-.204
0.83
NEUTS
5543.500
15554.500
-4.977
0.0006
Describing continuous data
• If the data is normally distributed
– Mean and standard deviation
• If the data is skewed or non-normally distributed
or is from a small sample (N<20)
– Median and range
Comparison of several means
• Data sets comprising more than two groups are common,
and their analysis often involves the comparison of the
means for the component subgroups. It is obviously possible
to compare each pair of groups using t tests, but this is not a
good approach. It is far better to use a single analysis that
enables us to look at all the data in one go, and the method
of choice is called analysis of variance
• If the data are not normally distributed or have different
variances, a non-parametric equivalent to the analysis of
variance can be used, and is known as the Kruskal-Wallis test
Paired data
• When we have more than one group of
observations it is vital to distinguish the case where
the data are paired from that where the groups are
independent. Paired data arise when the same
individuals are studied more than once, usually in
different circumstances. Also, when we have two
different groups of subjects who have been
individually matched, for example in a matched
pair case-control study, then we should treat the
data as paired.
• A one sample t test is used to examine the data. The
value t is calculated from
– t = sample mean - hypothesised mean
standard error of sample mean
• In a paired analysis where one set of observations are
subtracted from the other set, the hypothesised mean
is zero. Thus the calculation of the t statistic reduces
to
– t = sample mean / standard error of sample mean
• The non-parametric equivalent to this test is the
Wilcoxon matched pairs signed rank sum test
A1
A2
A3
A5
A6
A8
A10
I1
I2
I5
Ratio AZU1/GAPDH
Samples CD34
MNC
2273 0.0379 0.4328
3667 0.0007 0.1021
2943 0.0003 0.0007
2334
0.014 0.0226
1759 0.0696 0.5604
2164 0.0349 0.3249
3022a 0.159 0.2487
1503
1684
1615
0.6225 0.9253
0.0647 1.4268
0.2571 0.2516
Wilcoxon Signed Ranks Test
Ranks
N
MNC - CD34
Negative Ranks
Pos itive Ranks
Ties
Total
1a
9b
0c
10
Mean Rank
2.00
5.89
Sum of Ranks
2.00
53.00
a. MNC < CD34
b. MNC > CD34
c. MNC = CD34
Test Statisticsb
Z
Asymp. Sig. (2-tailed)
MNC - CD34
-2.599 a
.009
a. Bas ed on negative ranks.
b. Wilcoxon Signed Ranks Tes t
Telomere length in Dyskeratosis Congenita
parent
-4.7163
-4.7163
-4.7163
-3.8798
-1.4062
-5.1662
-2.2144
-4.439
-4.2654
-4.9991
-0.5679
-5.3408
-1.2779
-1.2779
-4.1954
-0.1936
0.1764
-1.0408
1.0755
0.1737
-2.1199
-2.3117
1.2593
0.5821
0.5821
1.3701
1.3701
-1.469
-1.469
child
-6.6238
-5.9629
-6.1392
-0.6173
-2.2264
-5.4028
-3.6383
-2.0056
-6.7593
-3.8157
-2.5027
-6.4229
-5.1118
-3.9373
-5.9093
0.4262
-1.5093
2.4508
0.3898
1.4716
-0.4074
1.3426
1.6527
1.3701
-1.469
4.2772
-1.2765
-1.4892
0.8735
Paired Samples Statistics
Pair
1
CHILD
PARENT
Mean
-2.0335
-1.9032
N
29
29
Std. Deviation
3.13806
2.27896
Std. Error
Mean
.58272
.42319
Paired Samples Test
Paired Differences
Pair 1
CHILD - PARENT
Mean
-.1303
Std. Deviation
2.09654
Std. Error
Mean
.38932
95% Confidence
Interval of the
Difference
Lower
Upper
-.9278
.6672
t
-.335
df
28
Sig. (2-tailed)
.740
Comparison of groups : continuous data
• Paired on non-paired?
• If non-paired and normally distributed with
similar variances : T-test
• If non-paired non-normally distributed or with
non-similar variances or very small numbers :
Mann-Whitney test
• Paired data – paired t-test or Wilcoxon Signed
Ranks Test
Comparing Proportions
• Qualitative or categorical data is best presented in
the form of table, such that one variable defines the
rows, and the categories for the other variable define
the columns. Thus in a European study of ASCT for
HD, patient gender was compared between the UK
and Europe
• The data are arranged in a contingency table
• Individuals are assigned to the appropriate cell of
the contingency table according to their values for
the two variables
COUNTRYG * PSEX Crosstabulation
Count
COUNTRYG
Total
europe
uk
16
16
PSEX
Female
610
100
710
Male
828
160
988
Total
1454
260
1714
COUNTRYG * PSEXG Crosstabulation
PSEXG
COUNTRYG
europe
uk
Total
Count
% within COUNTRYG
% within PSEXG
Count
% within COUNTRYG
% within PSEXG
Count
% within COUNTRYG
% within PSEXG
1.00
828
57.6%
83.8%
160
61.5%
16.2%
988
58.2%
100.0%
2.00
610
42.4%
85.9%
100
38.5%
14.1%
710
41.8%
100.0%
Total
1438
100.0%
84.7%
260
100.0%
15.3%
1698
100.0%
100.0%
Chi-squared test (2)
• A chi-squared test (2) is used to test whether
there is an association between the row variable
and the column variable. When the table has
only two rows or two columns this is equivalent
to the comparison of proportions.
• The first step in interpreting contingency table
data is to calculate appropriate proportions or
percentages. The chi-squared test compares the
observed numbers in each of the four categories
and compares them with the numbers expected if
there were no difference between the distribution
of patient gender
• The greater the differences between the observed
and expected numbers, the larger the value of 2
and the less likely it is that the difference is due to
chance
COUNTRYG * PSEXG Crosstabulation
PSEXG
COUNTRYG
europe
uk
Total
Count
% within COUNTRYG
% within PSEXG
Count
% within COUNTRYG
% within PSEXG
Count
% within COUNTRYG
% within PSEXG
1.00
828
57.6%
83.8%
160
61.5%
16.2%
988
58.2%
100.0%
2.00
610
42.4%
85.9%
100
38.5%
14.1%
710
41.8%
100.0%
Total
1438
100.0%
84.7%
260
100.0%
15.3%
1698
100.0%
100.0%
Chi-Square Tests
Pearson Chi-Square
Continuity Correctiona
Likelihood Ratio
Fisher's Exact Test
Linear-by-Linear
Association
N of Valid Cases
Value
1.418b
1.260
1.428
1.417
df
1
1
1
1
Asymp. Sig.
(2-sided)
.234
.262
.232
Exact Sig.
(2-sided)
Exact Sig.
(1-sided)
.246
.131
.234
1698
a. Computed only for a 2x2 table
b. 0 cells (.0%) have expected count less than 5. The minimum expected count is
108.72.
Fisher’s Exact Test
• When the overall total of the table is less than 20, or
if it is between 20 and 40 with the smallest of the
four expected values is less than 5, then Fisher’s
Exact Test should be used.
Crosstab
TRMV
DISG
3.00
SURV
.00
1.00
Total
Count
% within SURV
% within TRMV
Count
% within SURV
% within TRMV
Count
% within SURV
% within TRMV
.00
1.00
15
100.0%
88.2%
2
66.7%
11.8%
17
94.4%
100.0%
1
33.3%
100.0%
1
5.6%
100.0%
Total
15
100.0%
83.3%
3
100.0%
16.7%
18
100.0%
100.0%
Chi-Square Tests
DISG
3.00
Pearson Chi-Square
Continuity Correctiona
Likelihood Ratio
Fisher's Exact Test
Linear-by-Linear
Association
N of Valid Cases
Value
5.294b
.847
3.905
5.000
df
1
1
1
1
Asymp. Sig.
(2-sided)
.021
.357
.048
Exact Sig.
(2-sided)
Exact Sig.
(1-sided)
.167
.167
.025
18
a. Computed only for a 2x2 table
b. 3 cells (75.0%) have expected count less than 5. The minimum expected count is .17.
• The chi-squared test can also be applied to larger
tables, generally called r x c tables, where r denotes
the number of rows in the table, and c the number of
columns.
• The standard chi-squared test for a 2 x c table is a
general test to assess whether there are differences
among the c proportions. When the categories in the
columns have a natural order, however, a more
sensitive test is to look for an increasing (or
decreasing) trend in the proportions over the columns.
This trend can be tested using the chi-squared test for
trend.
• In the table below the relation between frequency
of Cesarean section and maternal foot size is
presented
Cesarean
section
Shoe Size
<4
4
4.5
5
5.5
6
Total
Yes
5
7
6
7
8
10
43
No
17
28
36
41
46
140
308
• The standard chi-squared test of this 2 x 6 table
gives and a 2 value of 9.29, with 5 d.f., for
which P=0.098. Analysis of the data for trend
gives a 2trend = 8.02, with 1 d.f. (P=0.005). Thus
there is strong evidence of a linear trend in the
proportion of women giving birth by Cesarean
section in relation to shoe size. This relation is
not causal, but reflects that shoe size is a
convenient indicator of small pelvic size
Categorical data – comparing
proportions
• Studies where there are 2 groups and the total
number of patients > 40 : Chi-squared test
• Studies where there are 2 groups and the total
number of patients < 40 or if more than 40 and a
single cell has less than 5 : Fisher’s Exact Test
• Studies where there are more than 2 groups but not
ordered : - Chi-squared test
• Studies where there are more than 2 groups which are
ordered : - Chi-squared test for trend
Exploring the relationship between
two variables
• Three possible purposes :
– a.) assess association e.g. body weight and blood
pressure
– b.) prediction e.g. height and weight
– c.) assess agreement e.g. blood pressure
measurement
Correlation
• Method for investigating the linear association
between two continuous variables
• The association is measured by the correlation
coefficient
• A correlation between two variables shows that
they are associated but does not necessarily
imply a ‘cause and effect’ relationship
• A t test is used to test whether the correlation
coefficient obtained is significantly different
from zero, or in other words whether the
observed correlation could simply be due to
chance
• The significance level is a function of both the
size of the correlation coefficient and the
number of observations. A weak correlation
may therefore be statistically significant if
based on a large number of observations, while
a strong correlation may fail to achieve
significance if there are only a few observations
Correlations
BET2MG
OPG
CRP
NTX
Pears on Correlation
Sig. (2-tailed)
N
Pears on Correlation
Sig. (2-tailed)
N
Pears on Correlation
Sig. (2-tailed)
N
Pears on Correlation
Sig. (2-tailed)
N
BET2MG
1
.
121
-.393**
.000
121
.620**
.000
121
.395**
.000
121
OPG
-.393**
.000
121
1
.
121
-.220*
.015
121
-.465**
.000
121
**. Correlation is significant at the 0.01 level (2-tailed).
*. Correlation is significant at the 0.05 level (2-tailed).
CRP
.620**
.000
121
-.220*
.015
121
1
.
121
.152
.097
121
NTX
.395**
.000
121
-.465**
.000
121
.152
.097
121
1
121
300
P=0.015
200
CRP
100
0
0
OPG
10
20
210
P=<0.0001
180
150
120
90
60
CRP
30
0
0
BET2MG
10
20
30
Problems with correlation analyses
• Biological systems are multifactoral so a simple twoway correlation may not be a true reflection of
what is being observed
• Spurious correlations
110
100
90
80
70
60
50
40
30
2
FOOT SIZE
4
6
8
10
12
Assessing agreement
• Neither correlation nor linear regression are
appropriate
• There may be a very high correlation, but one
method gives a systematically higher/lower
reading
• Linear regression, the data is not independent
• The only appropriate way is to subtract one
observation from the other, and plot against an
index variable
Correlation between PCR and TAQman for measuring MRD
PCR
PCR
Pearson Correlation
1.000
TAQ
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
107
.739
0.0006
107
TAQ
1.000
107
.7
.6
.5
.4
.3
.2
PCR
.1
0.0
0.0
TAQ
.1
.2
.3
.4
.5
.6
.7
.2
.1
0.0
-.1
-.2
DIFFER
-.3
-.4
-.5
0
SAMPLE
20
40
60
80
100
120
Paired Samples Test
Paired Differences
PCR - TAQ
95% Confidence
Interval of the
Difference
Std. Error
Mean
Std. Deviation Mean
Lower
Upper
2.830E-02
8.117E-02 7.847E-03 1.274E-02 4.386E-02
t
3.606
df
106
Sig. (2-tailed)
0.0002
Linear regression
• Linear regression gives the equation of the straight
line that describes how the y variable increases (or
decreases) with an increase in the x variable. y is
commonly called the dependent variable, and x the
independent, or explanatory variable
• A t test is used to test whether the gradient b differs
significantly from a specified value (usually zero)
• Assumptions
– for any value of x, y must be normally distributed
– the magnitude of the scatter of the points about
the regression line is the same throughout the
length of the line
– the relation between the two variables should be
linear
22
20
18
TLENGTH
16
14
12
0
AGE
10
20
30
40
50
60
70
22
20
18
16
14
12
0
AGE
10
20
30
40
50
60
70
Coefficientsa
Model
1
(Cons tant)
AGE
Uns tandardized
Coefficients
B
Std. Error
17.893
.317
-.049
.010
a. Dependent Variable: TLENGTH
Standardized
Coefficients
Beta
-.462
t
56.390
-4.809
Sig.
.000
.000
4
3
2
1
0
-1
-2
-3
-4
0
AGE
10
20
30
40
50
60
70
Practical application
• Y = mx + c
• Telomere length = age * -0.049 + 17.89
• Substituting in the above equation for ages of 30
and 60
• 16.42 = 30*-0.049 +17.89
• 14.95 = 60*-0.049 +17.89
22
20
18
16
14
12
0
AGE
10
20
30
40
50
60
70
Survival data
• Has 2 components
• The event of interest and the time to the event
• Special statistical methods are required – it is not
appropriate to use tests for categorical data
Life Table Analysis
• Survival data are usually summarised as survival or
Kaplan-Meier curves
• Based on a series of conditional probabilities
• For example, the probability of a patient surviving 10
days after a transplant, is the probability of surviving
nine days, multiplied by the probability of surviving
the 10th day given that the patient survived the first
nine days.
Patient number
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Dead
Alive
0
40
80
120
160
200
Days post BMT
240
280
320
Table 1. Life table for fifteen patients who received an allogeneic stem cell transplant
Time (days)
Status
Number at risk
Probability of
survival
Standard error
16*
0
15
1.00
26
1
14
0.93
0.069
66
1
13
0.86
0.094
69*
0
12
74
1
11
0.78
0.113
82*
0
10
88
1
9
0.69
0.129
89*
0
8
117*
0
7
133*
0
6
144*
0
5
172*
0
4
252*
0
3
291*
0
2
305*
0
1
Probability %
100
80
60
40
20
0
0
50
100
150
200
250
Days post BMT
300
350
Outcomes suitable for KaplanMeier analyses
• Survival (event of interest is death, patients alive
are censored)
• Disease-free survival (events of interest are either
death or disease relapse, patients alive and in
remission are censored)
• Primary graft failure
• Acute graft versus host disease
Overall and leukaemia-free survival for 111 patients with CML
in CP allografted with stem cells from HLA-identical sibling donors
100
OS
Probability (%)
80
67%
60
45%
40
LFS
20
0
0
1
2
3
Years post BMT
4
5
HH/ICSM May 2003
Probability of graft failure (%)
Graft failure following BMT for 1stCP CML with a VUD
20
15
13.2 Gy (n=57)
10
9%
5
14.4 Gy (n=44)
0%
0
0
60
120
180
240
Days post BMT
300
360
ICSM/HH April 2003
Probability of CMV reactivation (%)
CMV reactivation following BMT with a VUD
effect of ganciclovir treatment
100
80
60
post infection treatment (n=72)
43%
40
35%
20
prophylactic treatment (n=49)
0
0
28 56 84 112 140 168 196 224 252 280 308 336 364
Days post BMT
ICSM/HH May 2003
Use of computers for data
collection/analysis
• Decide what data needs collecting (for statistical
purposes) and then try if appropriate design a
form (this is best done in a database, eg Microsoft
ACCESS)
• Get the computer to do as much of the work as
possible. ie calculation of ages, surface area etc
• Think ahead to what format the spreadsheet/stats
package requires the data to be in
• For analysis purposes, its much easier to work
with numbers and codes, as opposed to
descriptions ie instead of male/female or m/f,
use 1 or 2
• Use a ‘code’ to identify missing data, eg 999 or
something ‘unlikely’
• Check the data before analysis, get ‘descriptive
statistics’
• Use appropriate statistical methods
• Statistical packages - SPSS, BMDP, STATA,
Statgraphics, MINITAB, STATXACT,
GENSTAT, SAS
Presentation of results
• Where possible give actual P values rather than ranges
– ie P=0.041 rather than P<0.05
• If a P value is not significant give the actual value and
not just NS
– ie P=0.15 rather than P=NS
• When presenting data it may be more useful to present
confidence intervals rather than a P-value
– ie lens A was more durable than lens B by 2.4 days (P=0.03), it
might be more informative to write - lens A was more durable
than lens B by 2.4 days (95%CI 0.3-4.5days)
• It is not necessary to give test results
– ie t=33.5, 28 dof, P=0.0001
• If a continuous variable is normally distributed
present, as a description of the data, the mean
and standard deviation, if not normally
distributed, a median and range
• Don’t quote more significant figures than
necessary
– ie mean patient age 34.2550 (std dev 11.4337),
(std dev 11.4) will suffice
34.3
1.000
0.800
0.600
0.400
0.200
0.000
32DP210
xl-6
Cell Line
xl-5
xl-4
xl-3
xl-2
xl-1
32D
Adhesion to FN (A490 absorbance units)
1.200
.8
8
.6
.4
XL2
.2
0.0
N=
GROUP
36
36
Writing the statistics section in a paper
• If power calculations were used to calculate the
sample sizes, details should be given
– eg Based on samples sizes of x in each arm, we should have
been able to detect a difference of y given 80% power at a
significance level of 0.05.
• State which statistical tests were used (reference
obscure ones).
– eg in order to investigate the differences between the
groups, a t-test was used for continuous data, and a chisquared test for categorical data
• If applicable, state whether standard deviations
or standard errors are quoted
• State whether p-values are from one or twotailed tests
– eg all quoted p-values are two-tailed
• Not necessary to quote which stats package
was used
Suggested Reading Material
• Essentials of Medical Statistics
– Betty Kirkwood
• Practical Statistics for Medical Research
– Doug Altman
• Statistical Methods in Medical Research
– Armitage and Berry
Summary
• If at all possible - consult a statistician before
starting your study
• Get a feel of your data by plotting results - don’t
rely on descriptive statistics alone
• Use appropriate statistical tests, not those that
give the ‘best’ results