Download Mean - The Hotel Network

Can I Believe It?
Understanding Statistics in Published
Literature
Keira Robinson – MOH
Biostatistics Trainee
David Schmidt – HETI Rural and
Remote Portfolio
Agenda
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Welcome
Understanding the context
Data types
Presenting data
Common tests
Tricks and hints
Practice
Wrap up
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Understanding statistics
 Never consider statistics in isolation
 Consider the rest of the article
 Who was studied
 What was measured
 Why was that measure used
 Where was the study completed
 When was it done
 It is the author’s role to convince you that their results
can be believed!
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Types of Data
Examples of data – Table 1
Diamond et al. 2006
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Types of data
 Numeric
 Continuous (height, cholesterol)
 Discrete (number of floors in a building)
 Categorical
 Binary (yes/no, ie born in Australia?)
 Categorical (cancer type)
 Ordinal categorical (cancer stage)
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Histograms
Represents continuous variables
 Areas of the bars represent the
frequency (count) or percent
Indicates the distribution of the
data
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0
10
20
30
40
Measures of association
160
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170
180
Height in cm
190
200
210
Stem and leaf plot- heights
6*
6*
6*
6*
6*
6*
6*
6*
6*
7*
7*
7*
7*
7*
7*
11
2
3333333
44444444444
555555555555
66666666666666666666666
777777777777777777777777777777
8888888888888888
99999999999999999999999999999999
0000000000000000000000000
1111111111111111111
222222222222
333333
44
55
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60
80
Frequency
40
60
20
40
20
0
0
Frequency
Skewed Data
40
60
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80
diastolic
100
120
140
100
150
systolic
200
Salient features- the mean
 The average value:
Birthweight
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mean - mean 1978
2010
3000
3500
grams
grams
Salient features- the median
 The observation in the middle
 Example- newborn birth weights
 3100, 3100,3200,3300,3400,3500,3600,3650 g
- (3300+3400)/2 = 3350
 Not affected by extreme values
 Wastes information
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Salient features- the mean and median
Birthweight
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mean 2010
3356
grams
median2010
3350
grams
Mean and Median
Mean is preferable
Symmetric distributions mean ~ median
 Present the Mean
Skewed distributions
 Mean is pulled toward the ‘tail’
 Present the Median
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Mean and Median
Mean
Median
13.6
11.7
20
10
0
Number of people
30
40
Body fat
0
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10
20
Body fat (%)
30
40
Variability – Standard deviation and variance
 The average distance between the observations and the
mean
 Standard deviation :
 with original units , ie. 0.3 %
 Variance =
 With the original units squared
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Range
 Example, infant birth weight
 3100, 3100,3200,3300,3400,3500,3600,3650, 3800
 Range = (3100 to 3800) grams or 700 grams
 Interquartile range: the range between the first and 3rd
quartiles (Q1 and Q3)
 3100, 3100,3200,3300,3400,3500,3600,3650 , 3800
 IQR = (3200 to 3600) grams or 400 grams
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Presenting variability
Present standard deviation if the
mean is used
Present Interquartile range if the
median is used
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Graphics for Continuous Variables
outlier
75th percentile
(Q3)
Wt
80
Maximum in
Q3
100
120
 Boxplot :
IQR
Minimum
in Q1
40
60
Median
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25th percentile
(Q1)
Categorical Variables- table summaries
Sport
Frequency
Percent
Cumulative
Percent
Soccer
25
12.6
12.6
Football
37
18.7
31.3
Basketball
23
11.6
42.9
Swimming
22
11.1
54.0
Golf
19
9.6
63.6
Rugby
44
22.2
85.9
Cycling
11
5.6
91.4
Tennis
17
8.6
100.0
TOTAL
198
100.0
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Bar charts
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nn
is
te
cy
cl
in
g
y
ru
gb
lf
go
m
in
g
ll
sw
im
ba
sk
et
ba
ll
tb
a
fo
o
so
cc
er
0
10
20
30
 Relative frequency for a categorical or discrete variable
Bar chart vs Histogram
 Histogram
 For continuous variables
 The area represents the frequency
 Bars join together
 Bar chart
 For categorical variables
 The height represents the frequency
 The bars don’t join together
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Pie chart
 Areas of “slices” represent the frequency
tennis
soccer
football
cycling
rugby
basketball
golf
soccer
basketball
golf
cycling
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swimming
football
swimming
rugby
tennis
24
Precision
26
Presenting statistics
 Tables should need no further explanation
 Means
 No more than one decimal place more than the original
data
 Standard deviations may need an extra decimal place
 Percentages
 Not more than one decimal place (sometimes no decimal
place)
 Sample size <100, decimal places are not necessary
 If sample size <20, may need to report actual numbers
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Statistical Inference
Sampling
Population
Inference
Sampling
Sample
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Sampling, cont’d
• A statistic that is used as an estimate of the population
parameter.
• Example: average parity
Population
Mean
Sample
Mean
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Confidence intervals
 We are confident the true mean lies within a
range of values
 95% Confidence Interval: We are 95% confident
that the true mean lies within the range of
values
 If a study is repeated numerous times, we are
confident the mean would contain the true mean
95% of the time
 How does confidence interval change as the
sample size increases?
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Confidence intervals cont’d
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Hypothesis testing
 Is our sample of babies consistent with the Australian
population with a known mean birth weight of 3500
grams?
 Sample mean = 3800 grams, 95% CI of 3650 to 3950
grams
 3800 lies outside of this confidence interval range,
indicating our sample mean is higher than the true
Australian population
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Hypothesis testing
 State a null hypothesis:
 There is no difference between the sample mean and the
true mean: Ho = 3500
 Calculate a test statistic from the data t = 2.65
 Report the p-value = 0.012
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What is a p-value?
 The probability of obtaining the data, ie a mean weight
of 3800 grams or greater if the null hypothesis is true
 The smaller the p-value, the more evidence against the
null hypothesis
 < 0.0001 to 0.05 – evidence to reject the null hypothesis
(statistically significant difference)
 > 0.05 – evidence to accept the null hypothesis (not
statistically significant)
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Summary – Confidence intervals and p
values
 P –value: Indicates statistical significance
 Confidence interval: range of values for which we are
95% certain our true value lies
 Recommended to present confidence intervals where
possible
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Analysing Continuous Outcomes
37
T tests
 What are they used for?
 Analyse means
 Provide estimate of the difference in means between the
two groups and the 95% confidence interval of this
difference
 P-value – a measure of the evidence against the null
hypothesis of no difference between the two groups
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T tests- paired vs independent
 Paired:
 Outcome is measured on the same individual
 Eg: before and after, cross-over trial
 Pairs may be two different individuals who are matched on
factors like age, sex etc.
Patient
Baseline weight
(kg)
3 months weight
(kg)
1
85
82
2
76
73
3
102
98
4
110
108
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Paired T-tests
 Calculate the difference for each of the pairs
Patient
Baseline
weight (kg)
3 months
weight (kg)
Difference
(kg)
1
85
82
-3
2
76
73
-3
3
102
98
-4
4
110
100
-10
Mean
93
88
-5
 The mean weight at baseline was 93 kg and the mean
weight at 3 months was 88 kg. The weight at 3 months
was 5 kg less compared to the baseline weight 95% CI
(-3, 12)
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Paired T-tests
 There was no evidence that there was a significant
change in weight after 3 months (p value = 0.19)
 Assumptions
 Bell shaped curve with no outliers
 Assess shape by graphing the difference
 Use a histogram or stem and leaf plot
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Independent T tests
 Two groups that are unrelated
 Eg: weights of different groups of people
Weight (kg)
NW Public School
SW Public School
52
45
51
54
71
82
14
15
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Independent samples t-tests
 Same assumption as for paired t tests plus the
assumption of independence and equal variance
Mean
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NW Public
School
(weight
kg)
SW Public
School
(weight,
kg)
Difference
(weight,
kg)
52
45
7
51
54
3
71
82
11
72
61
11
62
61
1 (-22,24)
Interpretation –independent t tests
The mean weight in NW Public was 62 kg
and the mean weight in SW Public was 61
kg
The mean difference in weight between
the two schools was 1 kg (-22, 24)
 There was no evidence of a significant
difference in weight between the two
schools (p=0.92)
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One-way Analysis of Variance (ANOVA)
 What happens when there are more than two groups to
compare?
 Null hypothesis: means for all groups are approximately
equal
 No way to measure the difference in means between
more than two groups, so the variance between the
groups is analysed
 Can measure variance within a group as well as variance
between groups
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One-way ANOVA
 Comparing multiple groups
NW Public
School
NE Public
School
SW Public
School
SE Public
School
42
39
46
56
53
52
51
45
46
58
56
41
75
41
44
32
56
65
63
56
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Interpretations – One-way ANOVA
 There was evidence of a difference between the average
student weight between the four schools p<0.05
 There was evidence of no difference between the
average student weight between the four schools
p>0.05
 Not advised to compare all means against each other
because there is an increased chance of finding at least
1 result that is significant the more tests that are done
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Assumptions ANOVA
 Normality, - observations for all groups are normally
distributed,
 Variance in all groups are equal
 Independence – all groups are independent of each
other
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Extensions of one-way ANOVA
 Two way-ANOVA:
 Multiple factors to be considered. Eg school and type of
school (public/private)
 ANCOVA – Analysis of Covariance
 Tests group differences while adjusting for a continuous
variables (eg. age) and categorical variables
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Linear Regression
 Measures the association between two continuous
variables (weight and height)
 Or one continuous variable and several continuous
variables (mutliple linear regression)
 What is the relationship between height and weight?
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Scatter plot of weight and height
40
60
Wt
80
100
120
 Correlation between height and weight = 0.75
160
170
180
190
Ht
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200
210
Scatter plot of body fat and height
20
0
10
Bfat
30
40
 Correlation between body fat and height = -0.23
160
170
180
190
Ht
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200
210
Linear regression
 Fits a straight line to describe the relationship
 Assumes
1. Independence for each measure (each person)
2. Linearity (check with scatter plots)
3. Normality (check residuals with a graph)
 Residuals are the difference between the data point and the
regression line
4. Homscedasticity
 Variability in weight does not change as height changes, ie
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Multiple Linear Regression
 Extends the simple linear regression
 Adjusts for confounding variables
 Example: Does smoking while pregnant affect infant
birth weight?
 Outcome variable: infant birth weight
 Exposure variable: maternal smoking
 Covariates (other variables of interest):
 Sex of the baby, gestational age
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Confounding variables
 A variable (factor) associated with both the outcome and
exposure variables
 Gestational age is associated with both smoking
(exposure) and the outcome (birth weight)
 Confounders can be assessed by checking the correlation
between the variable of interest and the outcome
variable
 Correlation coefficient : -1.0 <r<1.0
 Rule of thumb: >0.5 or <-0.5 should be considered a
confounder
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Example of weight vs height adjusting for
sex
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Summary for continuous outcomes
 Comparing means from two group
 Use t- tests (paired for same person comparison,
independent for independent groups comparison)
 Comparing means for more than two groups
 One-way ANOVA
 Comparing means for two or more groups and adjusting
for other variables (ANCOVA)
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Summary for continuous outcomes
 Assessing the relationship between two continuous
variables
 Simple linear regression
 Assessing the relationship between two or more
variables
 Multiple linear regression
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Analysing Categorical Outcomes
59
Chi-square tests
What can a chi-square test
answer?
Chi-Square tests
 2x2 tables:

Smoking
Total
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Low birth weight (<2500
grams)
Total
<2500
grams
>2500
grams
No
5
100
105
Yes
25
75
100
30
175
205
Chi-square tests
 Can be used for paired (same person under two different
conditions) or independent samples (unrelated people in
different groups)
 Used often in case-control studies where the outcome is
categorical (or dichotomous)
 Tests no association between row and column factors
 Smoking and low birth weight association
 The study design defines the appropriate measure of
effect
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Cohort studies
 Exposure is determined by
 Randomisation to different groups
 followed over time
 Outcome is determined at the end of follow up
 Rate of outcome can be estimated
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Cohort studies continued
 Eg. Rate of low birth weight in:
 Smokers: rate = 25/100 = 0.25 = 25%
 Non-smokers: = 5/105 = 5%
 Relative risk (RR) = 25/5=5 times higher risk of low birth
rate in smokers relative to non-smokers
 Risk Difference (RD) = 25-5 = 20
 No relative difference between the low birth rate in
smokers and non-smokers RR =1.0
 No absolute difference in the low birth rate in smokers
and non-smokers = RD
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Cross-Sectional Studies
People observed at one point in time (questionnaire)
Exposure and outcome are measured at the same time
Causal associations cannot be deduced
Rate ratio (RR) = 25/5=5 times higher risk of low birth
rate in smokers relative to non-smokers
 Rate Difference (RD) = 25-5 = 20
 No relative difference between the low birth rate in
smokers and non-smokers RR =1.0
 No absolute difference in the low birth rate in smokers
and non-smokers = RD




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Case-control studies
 Use for rare outcomes (example: child prodigies)
 Children are selected based on being a prodigy
 Eg. 100 child prodigies and 100 children with normal
intelligence
 Determine exposure retrospectively
 Cannot obtain a rate
 Must obtain the odds of the outcome and compare
using an odds ratio
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Case-Control studies
Child prodigy
Fish oil
supplements
during
pregnancy
Total
Yes
No
No
30
50
105
Yes
70
50
100
100
100
205
Total
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Case-control studies
 Odds of being a prodigy:
 In exposed: 70/50 = 1.4
 In unexposed: 0.6
 Odds ratio:
 1.4/0.6 = 2.3
 2.3 times more likely to have a child prodigy if maternal fish
oil supplements were taken during pregnancy
 Null hypothesis
 No association between the exposure and the outcome
 Odds Ratio = 1
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Summary of RR and OR
 Both compare the relative likelihood of an outcome
between 2 groups
 RR=1 or OR = 1
 Outcome is as likely in the exposed and unexposed groups
 RR>1 or OR >1
 The outcome is more likely in the exposed group compared
to the unexposed group
 The exposure is a risk factor
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Summary of RR and OR
 RR<1 or OR<1
 The outcome is less likely in the exposed group compared
to the unexposed group
 The exposure is protective
 RR cannot be calculated for a case-control study
 OR ~ RR when the outcome is rare
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Extensions of Chi-square
 Small sample sizes
 Fisher’s exact test
 Recommended when n<20 or 20 <n<40 and the smallest
expected cell count is <5
 Paired data
 Exact binomial test for small sample sizes
 McNemar’s test
 Multiple regression:
 Logistic regression
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Non-parametric tests
Parametric test
Non parametric tests
Independent samples t-test
Wilcoxon-Mann-Whitney test
Paired t-test
Wilcoxon signed rank sum test
One-way ANOVA
Kruskal Wallis
Chi-square test
?
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Spurious statistics
Fact or Fiction
 Vaccines and autism?
 Cell phones and brain tumours?
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75
Common errors
 60.182 kg or 61kg?
 Reporting measurements with unnecessary precision
 Age divided into 20-44 years, 45-59 years, 60-74 years,
75+ years
 Dividing continuous data without explaining why or how
 Certain boundaries may be chosen to favour certain results
 Presenting Means and SD for non-normal data
 What should be presented instead?
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Common Errors
 “The effect of more exercise was significant”
 “The effect of 40 minutes of exercise per day was
statistically significant for decreasing weight (p<0.05)”
 “40 minutes of exercise per day lowered the mean
weight of the group from 95 kg to 89 kg, (95% CI = 75105 kg, p= 0.03)
 Checking the distribution of the data to determine the
appropriate statistical test
 Using parametric tests when data is not normal
 Using tests for independent data when the data is paired
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Common Errors
 Using linear regression without confirming linearity
 Not reporting what happened to all patients
 Leads to bias of the results
 Data dredging
 Multiple statistical comparisons until a significant result is
found
 Not accounting for the denominator or adjusting for
baseline
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Example
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Common Errors
 Selection Bias
 Sampling from a bag of candy where the larger candies are
more likely to be chosen
 On November 13, 2000, Newsweek published the following
poll results:
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Selection Bias
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Selection Bias
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Common Errors
 Other biases (measurement bias, intervention bias)
 Using cross sectional studies to infer causality
 More likely to have a c-section if attending a private
hospital instead of a public hospital
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Practical example
 Working in groups quickly read the article provided
 Summarise
 What data they used
 What test
 Do you believe their findings?
 Can you explain why?
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Summary
 Statistics must be understood in the context of the whole
article
 Statistical tests must fit the data type
 Findings should be presented appropriately
 Beware flashy stats!
 It’s the author’s job to justify their choices
 If you don’t believe it- can you base your practice on it?
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Questions?
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