Applied statistics
Katrin Jaedicke
katrin.jaedicke@ncl.ac.uk
What you will learn in this course
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Basic statistic terminology
Using SPSS
Summary statistics
Cross-sectional and longitudinal comparisons of 2 and more samples
Corrections for multiple comparisons
Correlations
Transformations
Creating graphs in SPSS and SigmaPlot
• To be confident in using statistics!
• The statistics presented in the lecture are correct (to the best of my
knowledge), but this does not imply that all other statistical methods
are wrong! (But be sure you know what you are doing if you are using
other methods!)
Introduction to SPSS
Comparison of 2 groups (k = 2)
independent samples
metric data categorical data
normal distribution
Shapiro-Wilk Test
no
yes
dependent samples
metric data
categorical data
normal distribution
Shapiro-Wilk Test
yes
no
Levene Test for
homogeneity of variances
yes
no
t-Test for
independent samples
(Student’s t-test)
Mann-Whitney U-Test
paired t-Test
Wilcoxon Test
Independent samples, dependent samples and replicates
a) Independent
samples
15 kg
b) Dependent
(related)
samples
c) Replicates
15 kg
5 kg
Starvation
15 kg
15.1 kg
10 kg
14.9 kg
15 kg
14.95 kg
Independent samples, dependent samples and replicates
Exercise Cell culture: Treatment 1
A
B
24 h later
D
0h
6h
Treatment 2
C
24 h
E
ELISA
Treatment 3
Metric and categorical data
Categorical
Metric
Age groups
Child
Teenager
Adult
Examples from the lab
Metric
Categorical
ELISA
Bradford protein assay
Cell proliferation
Flow cytometry
Realtime PCR
States of disease severity
Cancer classifications
Staining categories
Number of people
Normal distribution
Height of each person
• Very few very small people
• Many average height people
• Very few very tall people
The Null Hypothesis
• The question that you ask when doing a statistic test.
• It is important to know which question the test is asking in
order to understand the result!
What we test in statistics: How big is the mistake that I make if I
reject the Null Hypothesis? (e.g. if I say the Null Hypothesis is wrong)
The accepted mistake is (generally) set at 5 %
< 5 % *p < 0.05 (small mistake)
< 1 % **p < 0.01 (even smaller mistake)
< 0.1 % ***p < 0.001 (very small mistake!)
The normal distribution test
(Shapiro-Wilk test) asks the
following question:
Do our data follow a normal distribution?
Answer to that question:
No-> p < 0.05
Yes ->
p > 0.05 e.g. the hypothesis is right and our data follow a
normal distribution!
Homogeneity of variance
How spread out
are two different
samples?
Null Hypothesis Question:
Are the variances in both populations equal?
p > 0.05 = homogeneity of variance!
Null Hypothesis Question for any tests looking at
differences between groups:
There are no differences between the groups.?
p < 0.05 = there is a significant difference between
the groups
Comparison of more groups (k > 2)
independent samples
metric data
no
homogeneity of variances
categorical data
normal distribution
Shapiro-Wilk Test
yes
no
sphericity
Mauchly’s Test
Levene Test
yes
metric data
categorical data
normal distribution
Shapiro-Wilk Test
yes
dependent samples
no
yes
oneway ANOVA
Kruskal-Wallis
t-Test with Bonferroni
correction
U-Test with
Bonferroni
correction
no
repeated measurement
ANOVA
paired t-Test with
Bonferroni
correction
Friedman Test
Wilcoxon Test with
Bonferroni
correction
Mauchly’s Test of Sphericity
Patient
Numbers
0h
24 h
48 h
0 h-24 h
0 h-48h
24h-48h
P1
P2
P3
P4
P5
Note: if you want to know how to calculate Variance, check
here: http://www.wikihow.com/Calculate-Variance
Null hypothesis question: Is the variance between all
group differences the same?
p > 0.05 = homogeneity of variance (Sphericity)!
Post-hoc testing and the Bonferroni correction
Serum protein
300
***
***
200
*
100
0
control
Bonferroni Correction:
A
B
C
significance value
number of tests
p<0.05 -> new p value =
p<0.01 -> new p value =
0.05
= 0.01
5
0.01
p<0.001 -> new p value =
5
= 0.002
0.001
= 0.0002
5
5 Student’s t-Tests:
1. Control-A
2. Control-B
3. Control-C
4. A-C
5. B-C
Error of Multiple testing ->
Control and C are
replicates!
Very small new p-values, risk of loosing all
significance, especially if small sample size.
Bonferroni-Holm or Benjamini-Hochberg
(Benjamini only parametric data)
correction: stepwise correction (less
conservative, more powerful)
Corrections for multiple comparisons (Bonferroni corrections)
ELISA
Replicates!
• As post-hoc testing, we do 5 comparisons which give us 5 different p values
• It does not matter if we have used (for each of the 5 tests, do not! mix different
tests!) Student’s t-test, the paired samples t-test, Mann-Whitney or the Wilcoxon
test to get these -> corrections should be done no matter which branch/side of the
overview diagram you are on
1.
2.
3.
4.
5.
Control-A (p= 0.0002)
Control-B (p= 0.003)
Control-C (p= 0.01)
A-C (p= 0.04)
B-C (p = 0.06)
The exact same Control data are used
3 times->Replicates!
The exact same stimulation data C are
used 3 times->Replicates!
We need to correct for the Error of Multiple testing e.g.
for the mistake of using Replicates!
Exercise Bonferroni-Holm
1. Put all the p values from the smallest to the highest into the K column
- 0.0002; 0.003; 0.01; 0.04; 0.06
2.
Use the new p values to define the level of significance (**)
Note:
If less tests are done (e.g. 3 or 4) or if more tests are done (e.g. 6, 7…), delete or
add cells in the excel spreadsheet and change K accordingly.
Transformations -> achieve parametric testing
Number of people
- To get not normal distributed data into a normal distribution
- To get data which does not have equal variances into data which has equal
variances
- After transformations, data have to be checked again for normal distribution
and equality of variance
- !use the new data for statistics, but not for graphs! Graphs should be done
with the original, untransformed data
Logarithm (log)
Square root (√)
Invert (1/x)
Height of each person
Correlations
metric data
categorical data
normal distribution
Shapiro-Wilk Test
yes
no
small sample size
no
Pearson correlation
yes
Spearman’s rank correlation
Correlations + Chi square
Correlations
- p<0.05 correlation significant -> draw line
- Correlation coefficient between 0 and 1
- < 0.3 weak correlation
- > 0.75 strong correlation
Chi square
-
Only Yes-No answers exist
For example: comparison of gender, races,
blood groups…
Important to test if patient groups are matched
The “grey” areas of statistics
Q: How important is the normal distribution?
A: The “big” tests such as ANOVA and repeated measures ANOVA,
but also the t-tests for larger sample sizes, can “cope” with having
only approximate normal distribution.
Q: How important is the equality of variance?
A: Very! A violation of equality of variances potentially changes test
results and may also reduce statistical power.
Q: What is a small and what is a large sample size?
A: There is no “definition” of small and large sample size, it depends
on the field of research what is commonly used. Rule of thumb:
sample size of n=4 is the minimum when I can do parametric
testing, anything less should be tested non-parametric.
Q: Do I always have to correct for multiple comparisons?
A: No, but you have stronger results if your p-values are still
significant after correction and they are less likely being open to
criticism of being a “chance” finding.
Mean and Median
Mean-> Normal distributed data
Add all numbers of analysed samples together and divide by n (sample size)
For example: 1, 2, 4, 6, 12
1+2+4+6+12=25
Mean: 25/5=5
Median-> Data are not normal distributed
Find the middle number of the analysed samples
For example:
Odd amount of numbers: 3, 9, 15, 17, 44
Middle number
Median: 15
Even amount of numbers: 3, 6, 8, 12, 17, 44
Add the 2 middles numbers and divide by 2
Median: (8+12)/2=10
Standard deviation, Standard error and Interquartile range
Standard deviation and Standard error-> Normal distributed data
Standard deviation: how much variation is there around the mean
- Small Standard deviation: data points are spread closely around the mean
- Large Standard deviation: data points are spread widely around the mean
- In Excel: =STDEV
Standard error: Standard deviation of the error of how accurate the mean is
-> does not add valuable information to the data, do not use!
Interquartile range-> Data are not normal distributed
first quartile (Q1) or lower quartile: 25th percentile
second quartile (Q2) or median: 50th percentile
third quartile (Q3) or upper quartile: 75th percentile
Interquartile range: Q3-Q1
Box plot