P-values and statistical inference Dr. Omar Aljadaan

The change arises as standard error is now calculated under the assumption that the two groups have the same population standard deviation, σ. This implies that both s 1 and s 2 are estimating the same quantity and so these are combined into the so called pooled estimation, S pooled as in the following table.

DistributionParameterSample estimateStandard error NormalMean difference µ 1 -µ 2 BinomialProportion

Z-test Z-test is equal to the estimate of the difference between treatments divided by the standard error of that difference. After calculating the z α we use the Standard Normal Distribution to figure out the α value, after that we can decide the confidence interval using the general form of it 100(1-α)%. The interval just include the null hypothesis value of zero difference. The value of α is termed as P-value

P-value can be interpreted as the probability of obtaining the observed difference, or more extreme, if null hypothesis is true.

Example N=148 patients, gone under pulmonary rehabilitation program. n int =93, m 1 = mean of X int =211m, SD(X int )=118 n cont =91, m 2 = mean of X cont =123m, SD(X cont )=99 Null hypothesis is d= m 1 –m 2 =0. Difference d= m 1 –m 2 = =88m SE Pooled = 16.1m Calculating d/SE pooled =88/16.1=5.465 > 1 more than 5 times the standard deviation of this distribution, from the null hypothesis.

This is very extreme observation and very unlikely to arise by chance since 95% of observation sampled from Normal distribution It is very unlikely that the measurements come from a Normal distribution whose mean is 0. So the notion of equality of effect of the two treatments suggested by the null hypothesis is rejected. The conclusion is that the intervention results in further distances walked in patients with chronic obstructive pulmonary disease than control management

P-value Example Change in blood pressure after exercise N=16 d, was calculated for each patient. If the Null hypothesis is true then the 16’d should be close to zero.

Using the table T1 with z=4.44, a P-value < is obtained We reject the null hypothesis.

Statistical inference Hypothesis testing is a method of deciding whether the data consistent with the null hypothesis. The calculation of the P-value is an important part of the procedure. If the p-value is less than or equal to α conclude that the data are not consistent with the null hypothesis. Whereas if the P- value is greater than α, do not reject the null hypothesis and view it as ‘not yet disproven’

P-value Small < αLarge ≥ α Your results are unlikely when the null hypothesis is true Your results are likely when the null hypothesis is true

We say our results are statistically significant if the p-value is less than the significant level α, usually Result isP-value ≤ 0.05P-value > 0.05 Statistically significantNot statistically significant DecideThat there is sufficient evidence to reject the null hypothesis and accept the alternative That there is insufficient evidence to reject the null hypothesis We cannot say the null hypothesis is true, only that there is not enough evidence to reject it

Confidence interval rather than p- value