### Statistical hypotheses

**Null hypothesis (H0):**No difference or relation exists; e.g. Treatment A is not better than Treatment B**Alternative or research hypothesis (H1):**Some difference or relation exists, e.g. Treatment A is better than Treatment B

### Statistical errors

**Type I error (alpha):**False positive (Falsely rejecting Null-hypothesis; i.e. null hypothesis is actually true but rejected)**Type II error (beta):**False negative (Falsely accepting Null-hypothesis; i.e. null hypothesis is actually false but not rejected)

You can never prove alternate hyopthesis, but you can reject the null hypothesis.

Mnemonic:

Alpha error (Type I)→ false Positive error → The P in Positive has one (I) vertical line, so corresponds to type I

Beta error (Type II)→ false Negative error → The N in negative has two (II) vertical lines, so corresponds to type II

Alpha and P-value

### P value: Probability of type I error

**Alpha:** Significance level – maximum tolerable probability of type I error

- a significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference.

Alpha sets the standard for how extreme the data must be before we can reject the null hypothesis. The p-value indicates how extreme the data are. We compare the p-value with the alpha to determine whether the observed data are statistically significantly different from the null hypothesis:

a. **If P-value < or = alpha:** Null hypothesis rejected (result is statistically significant)

b. **If P-value > alpha:** Null hypothesis not rejected (result is statistically non-significant)

**Mnemonic:**

When assessing **P-Value vs. Alpha**:

a. If P(enis) is small (i.e. P-value < Alpha) –> REJECT (the null (nude))

b. If P(enis) is big (i.e. P-value > Alpha) –> ACCEPT (the null (nude))

P-value is the probability that the observed difference between groups is due to chance (random sampling error) when null hypothesis is true. So, if my p-value is less than alpha (<0.05), then there is less than 5% probability that the null-hypothesis is truer.

### Beta and Power

**Beta:** Probability of type II error

**Power:** Probability of correctly rejecting null hypothesis (true positive)

- Power = 1 – Beta
- Statistical power can be increased by: increasing sample size, reducing beta and increasing sensitivity
- By convention, most studies aim to achieve 80% statistical power

**4 Inter-related features of Power: ****Mnemonic:** BEAN

1. **B**eta error: As beta increases, power decreases

2. **E**ffect size: As effect size increases, power increases

3. **A**lpha error: As alpha error increases, power increases

4. **N** (Sample size): As sample size increases, power increases

### Confidence interval and confidence level

**Confidence interval (CI):** Interval within which a parameter value is expected to lie with certain confidence levels (1-alpha), as could be revealed by repeated samples.

- CI for sample mean = Mean +/- z (Standard error or SE)
- SE = Standard deviation or SD/√sample size or n
- Larger thhe sample size, narrower is CI
- Smaller the SD, narrower is CI
- There is no way to achieve 100% confidence
- Confidence limits are the upper and lower boundaries of confidence intervals

Corresponding to alpha = 0.5, 95% CI is often used.

- For 95% CI, Z = 1.96.
- For 99% CI, Z = 2.58.

**Interpretation**

- Overlapping confidence intervals between two groups signify that there is no statistically significant difference.
- Non-overlapping confidence intervals between two groups signify that there is a statistically significant difference.
- If the confidence interval includes the null hypothesis, the result is not significant and the null hypothesis cannot be rejected.
- If the 95% confidence interval of relative risk or odds ratio includes 1.0, the result is not significant and the null hypothesis cannot be rejected.
- If the 95% confidence interval of a difference between the means of two variables includes 0, the result is not significant and the null hypothesis cannot be rejected.

- A 95% confidence interval that does not include the null hypothesis corresponds to a p-value of 0.05
- A 99% confidence interval that does not include the null hypothesis corresponds to a p-value of 0.01