Statistical Inference, Hypothesis testing and Estimation

1
BITS PIECES From PREVIOUS LECTURES
2
T-TESTS
T-tests are used to compare the mean of the same
CONTINUOUS variable in 2 or more groups.
3
CHI - SQUARED TESTS
Chi-Squared tests are used to check whether or
not two CATEGORICAL variables are associated,
i.e., if they are not associated then they are
independent.
As we are working with categorical variables the
data for the two variables is summarised in a
contingency table (a crosstabulation in SPSS).
Under the assumption of independence (no
association between the 2 variables) the expected
frequencies for each cell are easily calculated.
4
Chi-squared test for association
H0 there is no association between 2 categorical
variables

• Continuity Correction
• for 2 x 2 tables
• Fishers Exact Test
• if greater than 20 of expected values are less
  than 5 (calculated for 2 x 2 tables only)
• any single cell 1 or less
• Pearson
• for tables that have more than 2 rows or columns.
• Mantel-Haenszel Test for trend (Chi square test
  for trend)
• When one of the variables is ordinal

5
Independent Groups Example in Lecture 5
H0 The variables smoking status and gender are
independent
P 0.376 (continuity corrected chi-square test,
SPSS)
DO NOT REJECT H0
6
Related Groups 1 Example in Lecture 5
H0 Endometrial ablation has no effect on
symptoms of women.
Pairs of reports of symptoms of discomfort for
each woman, one before the operation (pre op) and
the other after operation (post op)
P 0.291 (McNemar test, SPSS)
DO NOT REJECT H0
7
Outbreak of influenza A (H3N2) in a
highly-vaccinated religious community a
retrospective cohort study. Nicholls S et al.
Communicable Disease and Public Health Dec 2004
7(4)272-277
The rate of influenza was significantly
associated with the age of the subject (p0.04).
Test used is (Chi square test for trend). From
the percentages the percentage of each age group
with the disease decreases with increasing age.
Chi-squared test for linear trend ?2 4.0 (df
1), P 0.04
8

• State the research question of interest,
• Summarise information in abstract in an
  appropriate contingency table,
• (Calculate expected frequencies for each cell),
• Write down the null and alternative hypothesis
  which you think are associated with the given
  P-value.
• What are the conclusions?

9
Communicable disease and Public Health (Dec
2004) Research Question within US drug
users Does area of residence affect take up of
Hepatitis B vaccination (HBV)?
Contingency table with respective proportions
(by row) and expected frequencies.
Red text shows expected count
10
H0 there is no association between area and HBV
H1 there an association between area and HBV
?2 test statistic 17.754 (calculated in
SPSS)
( 140 - 116.9 0.5)2 116.9 (22.6)2
/116.9 4.38 5.11 3.81 4.45 17.75
REJECT H0
P value P (?2 ? 17.754) lt 0.01 ? lt0.05
There a significant association between area and
the take up of Hepatitis B Vaccine
11
Difference in proportion taking up HBV is 0.199
or 20 higher take up of HBV in Bronx 95 CI for
(p1 p2) where p1 and p2 are proportions (p1
p2) ? 1.96 x

(0.110, 0.288) This 95 CI excludes 0 11 to
29 higher take up of HBV in Bronx
12
Non-parametric methods

• Mr Gordon Prescott

13
Six main tests
14
Non-parametric methods

• Many of the statistical methods encountered so
  far require certain assumptions about the data
• These tests may give misleading results if their
  assumptions do not hold
• When the data does not follow certain
  distributional assumptions
• use non-parametric methods
• transform the data
• ...

15
Parametric v non-parametric tests

• Parametric tests (eg t-tests) assume that the
  data follows a particular distribution (eg Normal
  distribution)
• Non-parametric tests do not assume a particular
  distribution of the data
• These methods are distribution free because they
  are based on the analysis of ranks and not the
  actual values
• The averages tested are usually the medians

16
Non-parametric methods

• Advantages
• no parametric assumptions about underlying
  distribution required
• can be used on ranked data
• mathematical concepts are simpler than for
  parametric tests
• Disadvantages
• less discriminating(less powerful)
• although simple, arithmetic can be lengthy
• do not easily provide magnitude of differences

17
Introduction to ranking

• 29 birthweights
• The smallest actual value is given the rank of 1,
  the largest is given the rank of 29
• If there are some values which occur more than
  once, then add up their ranks for these values
  and divide by the number of observations with the
  same value

18
Example
Birthweight Rank 2.34 1 1 2.38
2 2.5 2.38 3 2.5 3.5 4 4 3.8 5 5 4
.0 6 7 4.0 7 7 4.0 8 7
Try exercise in pg 43 (bottom)
19
Distribution of birthweights
20
Class Exercise pg 43
21
When to use non-parametric tests?

• Data has been measured on an ordinal scale,
  (suggest at least 5 ordered categories)

• Rank ordered data (e.g. placing in race 3rd,
  7th),

• Small sample sizes (consider if nlt30, but
  not absolute rule),

• Continuous/discrete data which do not follow a
• Normal distribution,

• Unequal variances across groups,

• Data with outliers.

22
Analysis of continuous data

• Comparison of two related groups
• Paired t-test
• Assumption that the differences came from a
  population following a Normal distribution
• Comparison of two independent groups
• Independent groups t-test
• Assumption that data came from populations
  following a Normal distribution

23
Comparison of two related groups

• Wilcoxon matched pairs test
• Data is continuous (interval) but assumptions of
  paired t-test are not satisfied
• Data is ordinal (ranked scale)
• H0 No tendency for the first outcome to be
  higher or lower than the second

24
Example Wilcoxon matched pairs test

• A crossover trial of pronethalol versus placebo
  for the prevention of angina was carried out.
  The outcome of interest was the number of angina
  attacks experienced. Twelve patients took part
  in the trial

H0 no tendency for the number of angina attacks
when on placebo to be higher or lower than when
on the active drug.
H0 is essentially saying that the distribution of
the differences in the number of angina attacks
between the placebo and the active drug is
located around zero (no difference).
25
Example data Wilcoxon matched pairs
Ordered from smallest difference to largest
(ignoring positive or negative sign)
Number of angina attacks experienced
26
SPSSWilcoxon matched pairs test
67/116.09
27
SPSSWilcoxon matched pairs test

• Statistically significant result (p0.028)
• Patients tend to have fewer attacks whilst on the
  active drug than when taking the placebo
• Note the median of the differences is equal to 7
  
• So seven fewer attacks on average on active
  drug than on placebo

P-value
28
Comparison of two independent groups

• Mann-Whitney U test
• Data is continuous/discrete, but assumptions for
  the independent t-test are not satisfied
• Data is ordinal (ranked)
• H0 The distribution of the two populations is
  the same
• (ie the two distributions do not differ in
  location)

29
Example Mann-Whitney test

• Bicep skinfold thickness has been measured in
  patients with two different types of intestinal
  disease.
• Research question
• Is there a difference in the median skinfold
  thickness between the two groups of patients?

30
Example data

• Crohns disease Coeliac disease
• (n20) (n9)
• 1.8 2.8 4.2 6.2 1.8 3.8
• 2.2 3.2 4.4 6.6 2.0 4.2
• 2.4 3.6 4.8 7.0 2.0 5.4
• 2.5 3.8 5.6 10.0 2.0 7.6
• 2.8 4.0 6.0 10.4 3.0

(Group A)
(Group B)
31
Computation (order observations from smallest to
largest 1st to 29th)

• 1 2 3 4 5 6 7 8 ORDER
• 1.8 1.8 2.0 2.0 2.0 2.2 2.4 2.5 OBSERVED
  VALUE
• 1.5 1.5 4 4 4 6 7 8 RANK
• A B B B B A A A GROUP

32
SPSS output Mann-Whitney
33
SPSS output Mann-Whitney

• The difference in distribution of bicep skinfold
  thickness was not found to be statistically
  significant (P0.15)
• Therefore the null hypothesis of equal
  distributions can not be rejected

P-value
34
Presentation of data
35
Comparison of more than two independent groups

• Kruskal-Wallis Test for k independent groups (3
  or 4 or 5 independent groups)
• Non-parametric equivalent of one way analysis of
  variance (ANOVA)
• H0 There is no difference in the distribution of
  values across the three groups (in the
  population)

36
Example Kruskal-Wallis

• Randomised comparison of three treatments for
  children who suffer from frequent and severe
  migraine.
• 18 children randomised (6 per treatment group)
• Headache activity after treatment was expressed
  as a percentage of baseline data
• (100 indicates complete absence of headaches,
  negative values indicate an increase in headaches)

37
Treatment for headaches data
38
SPSS Kruskal-Wallis
39
SPSS Kruskal-Wallis

• P value 0.06
• Borderline significance (officially not
  significant)
• Close to cut off point for statistical
  significance
• Suggests that there may be a difference in the
  effectiveness of treatment

P-value
40
Multiple comparisons

• It is possible (as in ANOVA) to examine which
  groups differ from one another
• Unfortunately these tests are not available in
  SPSS
• Options (not examinable)
• Pairwise Mann-Whitney tests (with some adjustment
  for multiple testing)
• Dunn multiple comparison (can be computed by
  hand)

41
Summaryparametric v non-parametric tests

• If the assumptions for the parametric test are
  met by the data then the parametric test is more
  powerful - and should be used
• Using a parametric test when the assumptions are
  not fully met can result in serious errors

42
Choice of statistical method

• For every research question there are a number of
  options for the statistical analysis
• (eg independent t-test, or transform data and
  then independent t-test, or Mann-Whitney).
• The decision as to which statistical method to
  use remains with the researcher after exploring
  the suitability of each method.

43
Next Friday (10 Nov)

• Prepare two answers from the first exam paper in
  your handbook January 2003
• Question 2 and Question 4. You will need to look
  ahead in notes to do parts 4, 8 and 9 of Q4.
• Room to be announced next week. Please come at
  the correct time 9-10, 10-11 or 11-12.
• 9-10 in 3rd floor conference room 3.052.
• 10-12 room(s) not yet confirmed.
• We will be going through answers as a classroom
  exercise.
• Everyone gains more if each person has done the
  preparation.
• You will learn more if you write answers in
  sentences.
• If short of time write brief notes.