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Can I Believe It? Understanding Statistics in Published Literature

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

### Chi-square tests

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

- 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

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
- 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

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

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

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?

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