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Updated: Apr 5 2022

# Statistical Hypotheses and Error

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• Hypotheses
• Null hypothesis (H0)
• hypothesis of no difference
• e.g., there is no link between disease and risk factor
• Alternative hypothesis (H1)
• hypothesis of difference
• e.g., there is a link between disease and risk factor
• Type I Error (False Positive)
• Stating there is an association when none exits
• incorrectly rejecting null hypothesis
• α = probability of type I error
• p = probability that results as or more extreme than those of the study would be observed if the null hypothesis were true
• general rule of thumb is that statistical significance is reached if p < 0.05
• Type II Error (False Negative)
• Stating there is no effect when an effect exists
• incorrectly accepting null hypothesis
• β = probability of type II error
• Power (True Positive)
• Probability of correctly rejecting null hypothesis
• power = 1 - β
• Power depends on
• sample size
• increasing sample size increases power
• size of expected effect
• increasing effect size increases power
• True Negative
• Probability of correctly accepting null hypothesis
• Confidence Interval
• Range of values associated with a confidence level indicating the likelihood that the true population value of a parameter falls within that range
• usually done with 95% confidence interval (2 standard deviations from the mean)
• e.g., based on our study data, we are 95% confident that the average salary of a teacher lies between \$30,000-45,000/year
• Confidence interval is calculated from statistics generated from the studied data
• Smaller confidence intervals suggest better precision of the data
• Larger confidence intervals suggest less precision of the data
• If confidence intervals of 2 groups overlap, there is no statistically significant difference
• A Priori Versus Post Hoc Analysis
• A priori comparisons
• comparisons planned prior to data analysis
• planning dependent on knowledge researchers have prior to conducting statistical tests
• Post hoc analysis
• researcher decides additional comparisons to make after viewing data
• choices dependent on knowledge researchers have gained after conducting statistical tests
• e.g., a test is run that says there is a difference between groups A, B, and C
• post hoc analysis would involve comparing group A to group B, B to C, and A to C to see between which groups the difference lies
• one potential hazard is an increased likelihood of spurious statistical associations
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