Statistical Fridays

Statistical Fridays J C Horrow, MD, MSSTAT Clinical Professor, Anesthesiology Drexel University College of Medicine

Previous Session Review • Statistics are functions of the data • Useful statistics have known distributions • Statistical inference = estimation; testing • Tests seek to disprove a “null” hypothesis

Session Review • Tests involve a NULL hypothesis (H0) an ALTERNATIVE hypothesis (HA) • Try to disprove H0 • There are 4 steps in hypothesis testing

Null and Alternative Hypotheses • Together, they describe all possibilities • EXAMPLE: If (H0) : BP=0, then (HA) : BP0. • EXAMPLE: If (H0) : SBP 80, then (HA) : SBP< 80.

How to formulate H0 • GOAL: To DISPROVE H0 • EXAMPLE: If our goal is to show DVT rates with a new oral anticoagulant X are lower than those with warfarin, then: H0 : RX RW and HA : RX < RW • Put the “equals” sign in H0

4 steps of hypothesis testing • Identify the test statistic • State the null and alternative hypotheses • Identify the rejection region • State your conclusion ------------------------------------------------------------ Example: ALT measured 3-months after starting drug X.

Step 1: Identify the test statistic • A statistic is a function of the data • Examples: average, maximum, rank-sum • Pick a statistic with known distribution • Observations vary, so functions of the data also have variation • The mean, x-bar, is most often used • Distribution is N (,2/n) if n sufficiently large • EX: mean ALT for drug X and drug W

Step 2: State H0 and HA • State in terms of population parameter • Put “equals” signs in H0 • Be sure to cover all possibilities • Example: H0 : X - W = 0HA : X - W  0 • N.B.: “two-sided” hypothesis

Step 3: Identify the rejection region • If T.S. differs “enough” from value “under H0” then we reject H0. • How much is “enough”?  rejection region • EX: T.S. is (x-barX – x-barW)R.R. is |x-barX – x-barW| > X-W  z/2

The Normal Distribution Z=1.965 R.R. -3s -2s -s 0 s 2s 3s

Step 4: State your conclusion • If T.S. is outside R.R., reject H0 • If T.S. within R.R., “cannot reject H0” • we do not “accept H0” if TS within RRmay state: data consistent with H0 • What about HA?If we reject H0, “data consistent with HA” • Why? Can never prove H0: this cohort is one of many possible!

Step 4: State your conclusion • Example: (x-barX – x-barW) = 1.3 (xULN)and X-W= 0.62 • R.R. = X-W  z/2 = 0.62  1.965 = 1.2183 • T.S. lies outside R.R. • Conclude: reject H0, data consistent with different S-ALAT for Xi and W groups.

The Normal Distribution Z=1.965 R.R. Dx=2.1s -3s -2s -s 0 s 2s 3s

Worked Example Do the patients in the C-section cohort have initial systolic BPs that are too low, i.e., less than 85 mmHg? STEP #1: Identify the T.S. T.S. = x-barSBP-init

Worked Example Do the patients in the C-section cohort have initial systolic BPs that are too low, i.e., less than 85 mmHg? STEP #2: State the hypotheses: H0 :   85 HA :  < 85 Note: this is a “one-sided” test

Worked Example Do the patients in the C-section cohort have initial systolic BPs that are too low, i.e., less than 85 mmHg? STEP #3: Identify the rejection region R.R. = (x-barSBP-init – 85)/SBP-init < z R.R. = (80.25 – 85)/1.790 < -1.645

Worked Example Do the patients in the C-section cohort have initial systolic BPs that are too low, i.e., less than 85 mmHg? STEP #4: State your conclusion R.R. -2.65 < -1.645  outside R.R. We reject H0. Data are consistent with initial systolic BPs that are too low.

The Normal Distribution Z=1.645 R.R. Dx=-2.65s -3s -2s -s 0 s 2s 3s

Session Review • Tests involve a NULL hypothesis (H0) an ALTERNATIVE hypothesis (HA) • Try to disprove H0 • There are 4 steps in hypothesis testing • Identify the test statistic • State the null and alternative hypotheses • Identify the rejection region • State your conclusion

Session Homework Use the C-section data Determine whether or not the increase in SBP exceeds 20 mmHg. Hint: first, form paired differences, then perform all 4 steps in testing

Statistical Fridays