Revision

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Revision

• A hypothesis test is a formal way to determine if
  a difference/change is due to an underlying
  difference/change in the population or due to
  chance.
• Compare means from two independent
  groups independent t-test
• Compare whether the mean change (before vs after)
  is equal to zero paired t-test

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Hypothesis Test (1)
Paired samples t-test Compare the mean of the
differences from paired data (to zero) H0
?d 0 Vs H1 ?d ? 0
Test is based on data collected from a paired
data set. The statistics of interest are
mean difference and SDd sd
0
tn-1
Test statistic
SE( )
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Hypothesis Test (2)
Independent samples t-test Compare the
population means in two independent groups
H0 ?A ?B Vs H1 ?A ? ?B
H0 ?A ?B 0 Vs H1 ?A ?B ? 0
Test is based on data collected from two
independent samples of sizes nA and nB. The
statistics of interest are
means and and SDs sA
and sB
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0
( )
tn-2
Test statistic
SE( )
with n nA nB lt 30
NB For both tests, if ngt30 the test statistic
with t-distribution can be assumed normally
distributed with mean 0 and standard deviation 1,
N(0,1).
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Assumptions
The assumption for the paired t-test is that the
differences have an approximately Normal
distribution. The assumptions for the
independent groups t-test is that the samples
come from populations having Normal distributions
with the same variance.
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Six main tests
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Hypothesis testing part 2

• Gordon Prescott

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Comparison of means

• What if there are more than two groups?
• Compare each pair of groups? NO
• The appropriate statistical test for when there
  are more than two groups is Analysis of Variance
  (ANOVA)

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Research question

• Is there a relationship between smoking and body
  weight?
• Subjects smoking status is classified as
• smoker
• given up smoking
• never smoked
• We want to compare body weight between each of
  the three smoking status categories

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Null hypothesis

• The null hypothesis is that there is no
  difference in population means between the groups
• The alternative hypothesis that there is a
  difference in population means between the groups

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Assumptions of ANOVA

• The sample data should come from populations
  which follow a normal distribution
• The variances of these populations should be the
  same (homoscedasticity)

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Comparison of means after ANOVA

• ANOVA will only indicate whether the means differ
  or not, it will not inform you of where the
  differences can be found (e.g. non-smokers
  weight differs from smokers but non-smokers
  weight does not differ from ex-smokers)
• Multiple comparison procedures can be used to
  test where the differences lie
• e.g. Scheffes test

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Descriptive statistics
Three means being compared
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Levenes test
Check to see that the data in the three groups
are equally spread out or not.
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ANOVA table
Formal hypothesis test to see whether the three
means are all equal or significantly different to
each other.
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Multiple comparison test
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• SPSS practicals page 91
• Task 2c requires ANOVA
• Instructions on pages 92-93

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Recap

• What is hypothesis testing?
• Theory
• In practice
• Comparison of two independent means
• Comparison of two paired means
• Comparison of more than 2 independent means

Independent t-test
Paired t-test
ANOVA
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IMPORTANT

• Open Door Sessions (1.005)
• WEDNESDAYS FROM 2pm TO 4pm
• Can make other appointments
• Student Clinics (1.026)
• DAILY FROM 1pm TO 2pm

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Tomorrows Practical

• SPSS exercise in page 91 of handbook,
• SPSS data file as always in
• F\data\Public Health\
• Tasks 3 and 4 to be completed tomorrow.

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HOW TO Print handouts of lectures, Add
comments in SPSS output, Export SPSS output to
Word, Read important values from tables.
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Insert
New Text
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NB Delete ALL Notes BEFORE Exporting Output to
Word
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File
Export
?
choose where to save your word document and give
it a name (eg. output.doc).
?
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NB To this new word file you can add more text,
delete and cut existing text, as well as shrink
and expand the graphs.
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Assignment

• Data file assignment2006.sav
• and a word file icu assignment2006.doc with
  another copy of the instructions are in
• F\data\Public Health

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Assignment

• Consider how to describe the data
• Formulate the research questions you would like
  the data to answer
• If a statistical test is required, decide what is
  the most appropriate test and why
• Then go to the data and find the information
  required and carry out the statistical tests
• Extract important information and include in
  write up
• Do NOT include large amounts of unedited SPSS
  output

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Deadline

• The deadline for the assignment is
• 12 NOON, MONDAY 20th NOVEMBER
• You can hand it in earlier if you wish
• You are welcome to discuss this work with other
  people
• Please ensure that your report is your own work
  and is different to the reports of other students
• Copied or plagiarised reports will be referred to
  the Head of Department

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Comparing Proportions

• Used for summarising qualitative data
• Calculated for different categories
• The proportions being compared usually come from
  a crosstabulation of two categorical variables
• Sample proportions may be used to draw inferences
  about population proportions
• These inferences may be expressed as confidence
  intervals or used in hypothesis testing

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Comparison of proportions from two independent
groups

• Is there a difference in the population
  proportion of men that are current smokers and
  the population of women that are current smokers?
• Is there an association between gender and
  smoking status?

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Null and alternative hypothesis

• The null hypothesis is that there is no
  difference in the population proportions of males
  who smoke and females who smoke
• (or no association exists between gender and
  smoking status in the population)
• The alternative hypothesis is that there is a
  difference in the population proportions
• (an association exists between gender and smoking
  status in the population)

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Chi-squared test for association

• Continuity Correction
• (tables with 2 rows and 2 columns)
• Fishers Exact Test
• (tables with 2 rows and 2 columns)
• Pearson Chi-squared Test
• (tables that have more than 2 rows or columns)
• Chi square test for trend
• (when at least one of the variables is ordinal)

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Contingency table
Total of row
Total of column
Grand Total
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Probability of two independent events

• Recall
• P(A and B) P(A) x P(B)
• If two events are independent, the result of one
  event is not dependent on the result of the other
  event

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Probability of an event

• If independent then
• P(male and smoker)
• P(smoker) x P(male)
• 82 x 120
• 272 272
• 0.133

• Therefore if smoking and gender are independent
  of one another the probability that a person in
  the population will be a male and a smoker is
  0.133,
• so in a sample of 272 people we would expect to
  find
• 0.133 x 272 36.2 male smokers

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Expected frequencies

• If two events are independent to one another,
  the expected frequency of events will be equal
  to
• Expected count
• (row total x column total)/ grand total
• 82 x 120 / 272
• 36.2

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Observed frequencies (expected frequencies)
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Chi-squared test

• The test of the null hypothesis is based on the
  difference between the observed and expected
  frequencies
• Under the null hypothesis this test statistic
  follows the Chi-squared distribution
• The value of the test statistic is then compared
  with the appropriate Chi-squared distribution
  (first proposed by Pearson)

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Chi-Squared

• Another important distribution related to the
  normal is the Chi-squared distribution.
• It is used when investigating categorical data.
• Large positive values occur with very low
  probability.

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Chi-squared test

• The greater the differences between the observed
  and expected statistics, the larger the
  Chi-squared statistic is, the more evidence that
  the two variables are associated.
• Comparison of the observed Chi-squared statistic,
  with tabulated critical values, will determine
  whether the evidence of association is
  significant at a given significance level.

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Assumptions of Chi-squared test

• When sample sizes are small, the expected
  frequencies may be small. For the Chi-squared
  test to be valid, no more than 20 of the cells
  should have an expected frequency of less than 5
  and no cells should have an expected frequency of
  less than one.
• If this does not hold the alternative test is
• Fishers Exact test
• (note for usually only for 2X2 tables)

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SPSS output
Note that 33.3 of males and 27.6 of females
smoke
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SPSS output
P-value
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Hypothesis
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Confidence intervals

• The difference in proportions is approximately 6
  (remember33.3 - 27.6)
• 95 CI for this difference can be obtained and is
  6 (-5 to 17)
• Note that the null hypothesised value 0 is
  included in the 95 CI
• This indicates that the null hypothesis can NOT
  be rejected at the 5 significance level

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Larger contingency tables

• Have seen that the Chi-squared test can be used
  to test for a difference between two proportions
• The Chi-squared test can also be applied to
  larger contingency tables
• Example
• Association between smoking (smoker, ex-smoker,
  never smoked) and gender

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Counts (expected frequencies)
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SPSS output
P-value
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Chi-squared test for association

• 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)
• 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

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Conclusion

• Recall
• H0 No association exists between gender and
  smoking status (i.e. the variables are
  independent)
• H1 There is an association between gender and
  smoking status
• There is no evidence to suggest that there is an
  association between gender and smoking status

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Comparison of proportions from two related groups

• Matched case-control study was conducted to
  investigate risk factors for diarrhoea in
  children.
• Does endometrial ablation (a conservative
  alternative to hysterectomy) have an effect on
  the presence of pain symptoms in women?
  Discomfort was assessed before and 6 months after
  surgery.

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McNemar test

• When the proportions are related (paired) the Chi
  square test is no longer valid since the
  observations in the contingency table are not
  independent of one another.
• The appropriate statistical test to apply is the
  McNemar test

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McNemar test
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SPSS output
P-value
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Confidence intervals

• The difference in proportions and the 95 CI is
• 17 (5 to 28)
• Note that the null hypothesised value of 0 is not
  included in the 95 CI
• Therefore the null hypothesis can be rejected at
  the 5 significance level

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McNemar test

• 24 women had symptom at pre-op and at post op
• 10 women had no symptom pre-op and still had no
  symptom post op
• 26 women who had symptom pre-op no longer had the
  symptom post op compared to 18 women who did not
  have the symptom pre-op but did post-op

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McNemar test

• 78 women had assessment pre and post operatively
• P 0.291
• No evidence to reject the null hypothesis
• Surgical treatment has had no statistically
  significant impact on whether patient has symptom

P-value
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Hypothesis testing or estimation?

• Quantification of the results by simple estimates
  is an essential part of the analysis of data
• A single number (P value) cannot convey all the
  necessary information appropriate estimates and
  confidence intervals are required as well

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Example

• In a study of patients admitted to an
  otolaryngology ward, 140 with nose bleeds were
  compared to 113 controls with other conditions.
  Patients were interviewed about their alcohol
  consumption (McGarry, 1994).

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Data
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Discuss the statement made by the authors

• The proportion of non-drinkers in the patients
  with nose bleeds was similar to that in the
  controls (34 vs 35), but the proportion of
  regular drinkers was significantly higher (45 vs
  30), Plt0.025, ? test of proportions)

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Discussion points

• The authors appear to have tested each line of
  the 3x2 cross tabulation
• 3 hypothesis tests using the same data
• Increased chance of type I error
• They should have done one single Chi-squared test
  on all the data
• Since categories of drinking are ordered an
  alternative would been to have done a Chi-squared
  test for trend
• SPSS does both of these automatically when you
  ask for a Chi-squared test