Can I Believe It? Understanding Statistics in Published Literature

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Can I Believe It? Understanding Statistics in Published Literature. Keira Robinson – MOH Biostatistics Trainee David Schmidt – HETI Rural and Remote Portfolio. Agenda. Welcome Understanding the context Data types Presenting data Common tests Tricks and hints Practice Wrap up.

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### Can I Believe It?Understanding Statistics in Published Literature

Keira Robinson – MOH Biostatistics Trainee

David Schmidt – HETI Rural and Remote Portfolio

Agenda
• Welcome
• Understanding the context
• Data types
• Presenting data
• Common tests
• Tricks and hints
• Practice
• Wrap up
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!

### Types of Data

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)
Histograms
• Represents continuous variables
• Areas of the bars represent the frequency (count) or percent
• Indicates the distribution of the data
Stem and leaf plot- heights

6* 11

6* 2

6* 3333333

6* 44444444444

6* 555555555555

6* 66666666666666666666666

6* 777777777777777777777777777777

6* 8888888888888888

6* 99999999999999999999999999999999

7* 0000000000000000000000000

7* 1111111111111111111

7* 222222222222

7* 333333

7* 44

7* 55

Salient features- the mean
• The average value:
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
Mean and Median
• Mean is preferable
• Symmetric distributions mean ~ median
• Present the Mean
• Skewed distributions
• Mean is pulled toward the ‘tail’
• Present the Median
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
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
Presenting variability
• Present standard deviation if the mean is used
• Present Interquartile range if the median is used
Graphics for Continuous Variables
• Boxplot :

outlier

Maximum in

Q3

75th percentile (Q3)

IQR

Median

Minimum in Q1

25th percentile (Q1)

Bar charts
• 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
Pie chart
• Areas of “slices” represent the frequency
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

### Statistical Inference

Sampling

Inference

Sampling

Sampling, cont’d
• A statistic that is used as an estimate of the population parameter.
• Example: average parity

Population

Mean

Sample

Mean

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?
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
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
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)
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
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
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.
Paired T-tests
• Calculate the difference for each of the pairs
• 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)
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
Independent T tests
• Two groups that are unrelated
• Eg: weights of different groups of people
Independent samples t-tests
• Same assumption as for paired t tests plus the assumption of independence and equal variance
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)
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
One-way ANOVA
• Comparing multiple groups
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
Assumptions ANOVA
• Normality, - observations for all groups are normally distributed,
• Variance in all groups are equal
• Independence – all groups are independent of each other
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
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?
Scatter plot of weight and height
• Correlation between height and weight = 0.75
Scatter plot of body fat and height
• Correlation between body fat and height = -0.23
Linear regression
• Fits a straight line to describe the relationship
• Assumes
• Independence for each measure (each person)
• Linearity (check with scatter plots)
• Normality (check residuals with a graph)
• Residuals are the difference between the data point and the regression line
• Homscedasticity
• Variability in weight does not change as height changes, ie
Multiple Linear Regression
• Extends the simple linear regression
• 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
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
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)
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

### Chi-square tests

What can a chi-square test answer?

Chi-Square tests
• 2x2 tables:
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
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
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
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
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
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
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
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
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

### Spurious statistics

Fact or Fiction
• Vaccines and autism?
• Cell phones and brain tumours?
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?
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 = 75-105 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
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
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:
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
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?
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?