1. Biostatistics for the biomedical profession Lecture III
Biostatistics for the biomedical profession Lecture III
BIMM18
Karin Källen & Linda Hartman
September 2015

2. Today Repetition Lecture 1: summary measures and graphical methods
Lecture 2: Normal distribution, generalisation, confidence interval, reference interval, t-test, ANOVA
Paired samples t-test
Non-parametric tests
Mann-Whitney’s test
Kruskal-Wallis’ test
Wilcoxon signed rank test

3. Repetition – the normal distribution
The (perfect) normal distribution
The mean, median, and mode all have the same value
The curve is symmetric around the mean; the skew and kurtosis is 0
The curve approaches the X-axis asymptotically
Mean ± 1 SD covers 2∙34.1%=68.2% of data
Mean ± 2 SD covers 2∙47.5%=95% of data
Mean ± 3 SD covers 99.7% of data
Excercise: What is the proportion of babies who will have a head circumference between -1 SD to +1 SD (=z-score -1 to +1)?
68%

4. Output from SPSS: Histogram for birth weight (in the ’births’ data set).
Excercise:
Are the data normally distributed?
Decide the limits between which 95% of all birth weights will be found.
Compute a 95% confidence interval for the mean

5. Birth weights cont.

6. Birth weights cont.
Tests of Normality
Kolmogorov-Smirnova Shapiro-Wilk
Statistic df Sig. Statistic df Sig.
Birth_weight , ,200* , ,147
*. This is a lower bound of the true significance.
a. Lilliefors Significance Correction
Two different methods to test for normal distribution
The p-value tells the likelihood for normal distribution

7. Are the data normally distributed?
Mean: 3539g
SD : 542g
N : 262
Excercise:
Are the data normally distributed?
Decide the limits between which 95% of all birth weights will be found.
Clue: 95% of all data with lie between SD and SD.
Compute a 95% confidence interval for the mean
Clue: 95%CI: Mean +/ * SEM
SEM= s/√n
Yes, no reason to doubt
95% Reference interval:
Lower limit: *542≈ 2477g
Upper limit: *542≈ 4601g
SEM=542/ √262 ≈ 33.5
95% Confidence interval:
Lower limit: *33.5 ≈ 3473
Upper limit: *33.5 ≈ 3605
Mean with 95%CI: ( )

8. But…. What to do if data do not follow a normal distribution?
How to perform descriptive statitics
2. How to compare the results between two samples?

9. Output from SPSS: Maternal BMI (kg/m2)
Excercise: Could a normal distribution be assumed?

10. Maternal BMI, cont…

11. Maternal BMI, cont….
1. Excercise: Could a normal distribution be assumed?
2. Which measurements should be used to produce descriptive statistics?
3. Under which circumstances could it be possible to nevertheless compare the means with a t-test?
No!!!
Median, inter-quartile range, histogram,
box plot etc.
We will repeat and learn more about the SEM (standard error of the mean)

12. Repetition – Central Limit Theorem
Mean has approximately normal distribution if
no. of observations is large (faster if distribution is symmetric)
Independent observations
from the same distribution 
We could often use
normal distribution to
test difference in mean
– even if observations are not normal
10000 samples of mean values from dice-rolls
Based on
10 rolls
100 rolls

13. Repetition: Normal distribution
Mean
420
Std
400
Median
280
QL-QU
percentile
Excercise:
Symmetric or assymetric?
Mean or median to describe data?
Use the median!
Example from Björk, Praktisk statistik för medicin och hälsa

14. Normal distribution: Each mean based on samples with n=10 N=10 N=20
Original dataset
Mean
420
Std
400
Median
280
QL-QU
percentile
Normal distribution:
Each mean based on samples with n=10
N=10
N=20
10 observations of CB-153 was sampled 1000 times and the mean (of the 10 observations) was calculated.
Histogram of the 1000 means.
N=50
N=100
The mean ( 𝑋 ) will approach normal distribution as the number of observations increase

15. Exercise - CLT Histogram of 𝑋 𝑋 = 𝟏 𝟏𝟎 𝒊=𝟏 𝟏𝟎 𝑿 𝒊 …
Original dataset
Mean
420
Std
400
Median
280
QL-QU
percentile
Exercise - CLT
Histogram of 𝑋
𝑋 = 𝟏 𝟏𝟎 𝒊=𝟏 𝟏𝟎 𝑿 𝒊 …
𝑋 = 𝟏 𝟏𝟎𝟎 𝒊=𝟏 𝟏𝟎𝟎 𝑿 𝒊
N=10
N=20
Let m=mean(X), s= Std(X) in the population
Excercise:
If 𝑋 is based on N observations:
Calculate the expected
mean( 𝑋 ) (= mean (of means) )
std( 𝑋 ) ( = standard deviation (of means)=standard error of means=SEM )
in the 4 histograms.
N=50
N=100
SEM=s/√n

16. Normal distribution: Let m=mean(X), s= Std(X) in the population
= 420
SEM
= 400/ 20
= 89
Mean( 𝑋 )
= 420
SEM
= 400/ 10
= 126
Original dataset
Mean
420
Std
400
Median
280
QL-QU
percentile
Let m=mean(X), s= Std(X) in the population
Excercise:
If 𝑋 is based on N observations:
Calculate the expected
mean( 𝑋 ) (= mean (of means) )
std( 𝑋 ) ( = standard deviation (of means)=standard error of means=SEM )
in the 4 histograms.
Mean( 𝑋 )
= 420
SEM
= 400/ 50
= 57
Mean( 𝑋 )
= 420
SEM
= 400/ 100
= 40
SEM=s/√n

17. Maternal BMI, cont….
1. Excercise: Could a normal distribution be assumed?
2. Under which circumstances could it be possible to nevertheless compare the means with a t-test?
No!!!
If the samples are large enough (at least approx ), we could nevertheless compare the means).

18. Repetition: Normal distribution Estimate of mean: Confidence interval
Hypothesis testing
Comparison of means:
Conf int for difference in means (2 groups)
T-test (2 groups)
ANOVA (> 2 groups)

19. Confidence interval
A confidence interval tells us within which interval the ’true’ estimate of a parameter probably lies-
E.g., a 95% confidence interval tells us between which limits the ’true’ estimate (with 95% certainty) lies.
Repetition: 95% of the data will lie between +/- 2 SD (1.96 exactly).
A 95% CI could be constructed (large samples):
(mean-1.96*SEM to mean+1.96*SEM)

20. Confidence interval Confidence grade 95% =100 %- 5%
Confidence grade 95% =100 %- 5%
i.e. 5% = 1/20 intervals (produced in the same way) will not cover the true value!
True value

21. Confidence intervals and reference intervals
A 95% confidence interval tells us between which limits the ’true’ estimate of the mean (with 95% certainty) lies:
(mean-1.96*SEM to mean+1.96*SEM)
A reference interval reflect the interval within which 95% of the population (or values) lies
Example:
Lower limit: mean – 1.96 * s Upper limit: mean * s

22. Confidence interval vs Reference interval
Confidence interval Interval for the mean of the population – with a specified confidence grade (here 95%)
95% CI: ( 𝑋 -1.96* 𝑠 𝑛 to 𝑋 +1.96* 𝑠 𝑛 )
Reference interval
Interval for the individual values of the population, with a specified coverage (here 95%)
95% ref int:
( 𝑋 -1.96*𝑠 to 𝑋 +1.96*𝑠)
The graphs are based on approx births.

23. The distribution of birth weight in two samples (n=100 and n=1000, respectively).
m=3477g, s=555g
m=3507g, s=580g
Excercise: The variance seems to be larger in the larger sample. Is that remarkable?

24. Exercise Which of the following statements are true? 1
Which of the following statements are true? 1
The larger the investigation, the narrower is the reference interval 2
The sample mean is always within the limits of the confidence interval 3
The population (true) mean is always within the limits of the confidence interval 4
The larger the investigation, the wider is the confidence interval 5
A confidence interval with 99% confidence grade is always wider than the corresponding confidence interval with 95% confidence grade
2 and 5 are correct

25. T-distribution
For a large sample 95% CI=( 𝑋 -1.96∙ 𝑠 𝑁 to 𝑋 ∙ 𝑠 𝑁 )
For small samples we must account for uncertainty in the estimate of the standard deviation 𝑠
95% CI=( 𝑋 -t(N-1)* 𝑠 𝑁 to 𝑋 +t(N-1)* 𝑠 𝑁 )
Degrees of freedom
t-constant for
95% CI 5
2.57 9
2.26 19
2.09 29
2.02 49
2.01 99
1.98 
1.96
N-1=”degrees of freedom”

26. The quantiles for T could be looked up in tables…
The quantiles for T could be looked up in tables…
But are rather produced by computer programs
As SPSS
Built-in in t-test & CIs

27. T-test for two independent samples (groups) - Example
Birth weight
Two groups
A: Smokers
B: Non-smokers

28. Repetition: Normal distribution Estimate of mean: Confidence interval
Hypothesis testing
Comparison of means:
T-test/Conf int for difference in means (2 groups)
ANOVA (> 2 groups)

29. T-test Assumptions The mean is a relevant summary measure
Independent observations
(e.g. no patient contributes more than one observation)
3. Observations are of Normal distribution OR
Both groups are large

30. T- test Example: Birth weight, Descriptives

31. Test procedure for t-test
Test variable:
D = Mean in group B - Mean in group A
H0: D = 0, Mean in group A = Mean in group B
H1: D  0, Mean in group A  Mean in group B
Construct a confidence interval for D
and/or
Calculate the p-value

32. Confidence interval - General formula
D is the observed average difference between the groups
D= 𝑋 𝐵 - 𝑋 𝐴
c is a constant that is dependent on the confidence level and the sample size
For 95% confidence level, and large sample, c ≈ 2
SE is standard error of D, a measure of how precise the estimated difference D is

33. T-test for two independent groups (cont.)
95% CI:
D= = 154 g
nA=183
nB=66
For now, we assume that the standard deviation is the same in both group A and B
 Base the analysis on a weighted (”pooled”) standard deviation sPooled

34. T-test for two independent groups (cont.)
sA=554
sB=478
535
535 ∙ =76
Use constant c for 95% confidence level (5% risk level) with
nA nB - 1 = 247 degrees of freedom  c  1.97
(obtained from statistical table for the t-distribution)

35.
T-test for two independent groups (cont.) The computer makes the calculations for us…. But we have to interpret the results!

36. Discuss:
D  c * SE  154  1.97 * 77  154  151
95% CI for the mean difference in birth weight is g
How do you interpret the confidence interval?
2. Is there a significant difference in Birth weight?
3. What can you say about the corresponding
p-value?

37. T-test for two independent groups Example of SPSS-output
Two different test versions depending on if equal standard deviation (variance) can be assumed or not
P-values for the t-tests
Levene’s test:
p-value (”Sig.”) testing H0: Variance in A = Variance in B
If not low (e.g. if p>.1) read from the upper row.
Difference with 95% CI

38. Presenting t-test results
Average in each group
Mean (possibly median as well for comparison)
± 95% CI sometimes relevant
± SE (i.e. 68% CI) not relevant
Variability in each group
Standard deviation
Percentiles if report space permits
Mean difference between the groups
± 95% CI usually relevant
P-value. Value of t-variable usually not relevant.

39. Elements of statistical inference
The p-value is the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis (H0) is true.
Type I error (often referred to as alpha) is the probability of rejecting H0 when in fact H0 is true.
Type II error (often referred to as beta) is the probability of accepting H0 when in fact H0 is false.

40. Important notes….
The p-value does not tell us anything about the size of the effect. Only how probable it is to obtain an effect of the size in our sample if the null hypothesis is true
The P-value is a function of both sample size and of true effect size
With large samples, statistically significant results could be found even if the size of the absolute effects are so small that they are of no clinical interest.
The fact that no significant results were found, does not mean that no difference exists. Perhaps the study had to low power to detect a true difference/effect/association.

41. Statistical significance vs. clinical relevance
Low p-value
How large is the difference?
Statistical significance:
”There is a difference”
Clinical relevance:
”Is the difference of importance?”
Effect estimation (CI) is needed!

42. Exercise – statistical inference
Answer:
E (or B) B C A D
Clinically relevant effect
No effect
95% Confidence intervals around the effect measure,
and p-values for the null hypothesis of ”no effect” in 5 investigations
Make pairs of the statements and study results (A-E) in the Figure 
1. Treatment effect cannot be detected, but cannot be ruled out
2. A clinically relevant effect is indicated, but is statistically uncertain
3. Treatment effect is statistically significant, uncertain if the effect is of clinical relevance
4. Clinically relevant effect that is statistically significant
5. Treatment effect is statistically significant, but a clinically relevant difference can be ruled out
Fr Jonas Björk: Praktisk statistik för medicin och hälsa

43. Excercise: Combine the statistical terms with the correct common phenomenons (in common language)
Type I error
Type II error
Confounding
Non-causal association
Mass-significance
Lack of power

44. More than two groups, one way ANOVA (analysis of variance)
Multiple T-tests could result in mass-significance!
Do ANOVA instead of repeated T-tests.
ANOVA:
H0: Mean1=mean2=mean3
H1: At least two of the means are different
In short: In an ANOVA, the total variance is devided into the within-groups, and between-groups variance.

45. ANOVA Compare variances Between groups (VB) Within groups (VW)
VB VW Ratio VB/VW
Large Small Large
Small Large Small
The quotient (F=VB/VW) is equal to 1 if group means are equal and >1 if they are not.
The corresponding test is called an F-test – and is based on the F-distribution

46. Example: ANOVA – to compare the birth weight between 4 parity groups

47. Example: ANOVA – to compare the birth weight between 4 parity groups

48. Example: ANOVA birth weight and parity, continued
Significance of the test:
Are all the means the same?
(m1=m2=m3=m4)
To check for pair-wise differences, post-hoc test could be performed

49. Post hoc tests Different methods to adjust for multiple comparison
Different methods to adjust for multiple comparison

50. Presenting ANOVA results
Average in each group
Mean (possibly median as well for comparison)
± 95% CI sometimes relevant
(± SE (i.e. 68% CI) not relevant)
Variability in each group
Standard deviation
Percentiles if report space permits
P-value together with
ANOVA-table if report space permits,
otherwise ”F(df1,df2)=…”

51. Paired samples & Non-parametric methods
Paired samples & Non-parametric methods

52. Difference between values
T-test for paired data
Preparation
ControlsDay2
SalDay2
Difference between values 11
915600
357800 2
953300
502200 3
650000
470000 4
700000
560000 5
736000 6
984000
556000 7
772000
418000 8
920000
600000 9
680000 10
520000
840000 12
533000
620000 13
510000
704000 14
722000
696000
Means
Two ways of comparing means:
Calculate the means of the groups, and estimate the difference
Estimate the difference for each row. Then calculate the mean of the differences

53. Difference between values
T-test for paired data
Preparation
ControlsDay2
SalDay2
Difference between values 11
915600
357800
557800 2
953300
502200
451100 3
650000
470000
180000 4
700000
560000
140000 5
736000
314000 6
984000
556000
428000 7
772000
418000
354000 8
920000
600000
320000 9
680000
400000 10
520000
840000
280000 12
533000
620000
-87000 13
510000
704000 14
722000
696000
26000
Means
824992,8571
570000

54. Difference between values
T-test for paired data
Preparation
ControlsDay2
SalDay2
Difference between values 11
915600
357800
557800 2
953300
502200
451100 3
650000
470000
180000 4
700000
560000
140000 5
736000
314000 6
984000
556000
428000 7
772000
418000
354000 8
920000
600000
320000 9
680000
400000 10
520000
840000
280000 12
533000
620000
-87000 13
510000
704000 14
722000
696000
26000
Means
824992,8571
570000
254992,9
Difference between means= mean of the differences

55. Difference between values
T-test for paired data
Preparation
ControlsDay2
SalDay2
Difference between values 11
915600
357800
557800 2
953300
502200
451100 3
650000
470000
180000 4
700000
560000
140000 5
736000
314000 6
984000
556000
428000 7
772000
418000
354000 8
920000
600000
320000 9
680000
400000 10
520000
840000
280000 12
533000
620000
-87000 13
510000
704000 14
722000
696000
26000
Means
824992,8571
570000
254992,9 s=
181454,0097
111808
216636,9
s (combined)
150709,3394
SEM
56962,77603
57898,64

56. Difference between values
T-test for paired data
Preparation
ControlsDay2
SalDay2
Difference between values 11
915600
357800
557800 2
953300
502200
451100 3
650000
470000
180000 4
700000
560000
140000 5
736000
314000 6
984000
556000
428000 7
772000
418000
354000 8
920000
600000
320000 9
680000
400000 10
520000
840000
280000 12
533000
620000
-87000 13
510000
704000 14
722000
696000
26000
Means
824992,8571
570000
254992,9
Thus, the mean is not influenced on whether the data are paired or not, but the estimate of the standard deviation is likely to differ with method.
Use analyses for paired data when adequate! s=
181454,0097
111808
216636,9
s (combined)
150709,3394
SEM
56962,77603
57898,64

57. Paired samples t-test Discuss:
Previous t-test was made to find differences between
independent groups of observations
Sometimes it is more powerful to test for differences within the same patient (or another paired measurement)
In a study of weight loss from spicy food, 12 subjects were weighed before and after a month on spicy food diet, see the table
Discuss:
How would you test if the diet gave weightloss?

58. Paired samples t-test 95% CI for mean(d): Mean(d)± t0.025(N-1)∙ 𝑠 𝑁
Do a paired samples t-test!
Calculate the differences di for each subject’s weights.
Test if mean(d) = 0
95% CI for mean(d):
Mean(d)± t0.025(N-1)∙ 𝑠 𝑁
Here:
-2.1 ± 2.2∙ =
-2.1 ± 1.9 = (-4.0,-0.17)
Discuss:
How do you interpret the CI?
Was the treatment effective

59. Paired samples t-test Paired test: 95% CI for weight loss (-4.0,-0.17)
If the researchers wouldn’t recognize the paired design, but did an independent groups’ t-test:
CI = (-18.6;22.76)
P=0.84
Why so wide?
Large variability BETWEEN subjects inflates the variability of the difference in an independent groups’ design!

60. Exercise: Creatinine was measured in 11 men and 12 women:
Men (nA = 11)
Women (nB = 12)
What test would you use to test if there is a difference?
T-test? Are the assumptions of the test met?

61. Parametric methods for group comparisons
Normally distributed outcomes/’large’ studies
Focus on mean comparisons
Two independent groups t-test
Paired groups (paired measurements) Paired t-test
> 2 groups Analysis of variance (ANOVA)
Regression analysis
What if assumptions are not met?
Non-parametric tests!

62. Comparison of medians – non-parametric tests
Comparison of medians – non-parametric tests

63. Non-parametric methods
Original measurements are converted to ranks in the analysis
H0: Distributions are equal in all groups
Median useful marker for differences in distribution
Insensitive to skewed distributions, extreme values
Can be used for ordinal data
E.g. 0 = No response, 1 = Mild response, 2 = Strong response

64. Difference between two independent groups: Mann-Whitney’s test
Rank the observations from the lowest to highest
Calculate rank sum in group A (WA) and in group B (WB)
 Straightforward generalization to more than two groups (Kruskal-Wallis test)
The larger the difference is in mean ranks WA/nA and WB/nB , the lower p-value will be
Mann-Whitney
Another name for the same test is ”Wilcoxon Rank sum test” which utilizes WA

65. Small Group Discussion...
Mann-Whitney…
Small Group Discussion...
Calculate the rank sum and the mean rank for males (and females if you have time)
For the group sizes nA = 11 (males) and nB = 12 (females), p < 0.05 if the rank sum for the smallest group (males) is below 100 or above 175
Conclusion?
How would you summarize the test?
Summarang (kvinnor) samt (män). Medelrang 9.8 respektive 15.7 (p ungefär 0.04)

66. Presenting Mann-Whitney results
Average in each group
Median (possibly mean as well for comparison)
Variability in each group
Percentiles or quartiles (in smaller groups) or min-max (in even smaller groups)
Standard deviation not relevant
Difference between the groups
P-value for M-W test
U-statistic sometimes relevant
Ideally: Median difference ± 95% CI sometimes relevant (could be calculated in e.g. SPSS)

67. Mann-Whitney Creatinine, cont

68. Extension to more than two groups Kruskal-Wallis test
Mann-Whitney U-test (k = 2)
E.g. H0: Distribution A = Distribution B
H1: Distribution A  Distribution B
Kruskal-Wallis test (k > 2 groups)
E.g. k = 3: H0: Distribution A = Distribution B = Distribution C
H1: Distribution A  Distribution B or
Distribution A  Distribution C or
Distribution B  Distribution C
Independent groups, independent observations within each group
Median useful marker for differences in distribution
The more the mean ranks differ, the lower the p-value will be

69. Non-parametric methods Paired samples
Non-parametric methods Paired samples

70. Non-parametric test for paired samples: Wilcoxon signed rank test
Sum the positive ranks:
=9.5
(Sum the negative ranks
=56.5 (=11*12/2-9.5) ) …
Spicy diet continued:
Subject
Pre
Post
Diff
Sign
Rank
Signed rank 1 65 62 -3 -
5,5
-5,5 2 88 86 -2 3
125
118 -7 11
-11 4
103
105 + 5 90 91 6 76 72 -4 8 -8 7 85 81
126
122 9 97 95 10
142
145
132 12
110 -5
-10
Subject
Pre
Post
Diff
Sign
Rank
Signed rank 1 65 62 -3 -
5,5
-5,5 2 88 86 -2 3
125
118 -7 11
-11 4
103
105 + 5 90 91 6 76 72 -4 8 -8 7 85 81
126
122 9 97 95 10
142
145
132 12
110 -5
-10

71. Wilcoxon signed rank test -Some remarks
Works for all types of distributions and for all study sizes
Effect can be summarized by the median of the paired differences together with 95% CI
Almost as powerful as the (paired) t-test if the differences are normally distributed
Power is decreased if many differences are zero

72. Comparison of different tests
Test-situation
Parametric test
Non-parametric test
Independent samples, 2 groups
T-test
Mann-Whitney
Independent samples, ≥ 2 groups
ANOVA
Kruskal-Wallis
Paired samples, 2 groups
Paired t-test
Wilcoxon rank sum test

73. Two broad categories of statistical methodsEx
Positive
Negative
Parametric methods
T-test
Results in both effect-measure (w CI) and p-value.
More effective to detect differences if data is (close to) normal
Based on assumptions about the distribution of data - typically 𝑁(𝜇,𝜎)
Test results could be sensitive to deviation from Normal distribution, especially in small studies
Non-parametric methods
Mann-Whitney
No assumptions about the distribution of data
useful also for data measured on an ordinal scale
Suitable for small studies
Less powerful than parametric methods (if normal distribution applies)
Typically results only in p-value (but sometimes an effect measure with CI could be computed)

74. Summary: Next lecture:
Repetition, Normal distribution, Reference interval, SEM, T-test, ANOVA
Paired t-test
Non-parametric methods
Mann-Whitney
Kruskal-Wallis
Wilcoxon signed rank test
Next lecture:
Subject
2x2 Table. Chi2-test, Fisher exact test
Probability, proportions
Linear regression
Correlation R2