Transcript document
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) : BP0. – 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 1.Identify the test statistic 2.State the null and alternative hypotheses 3.Identify the rejection region 4.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 – • Pick a statistic with known distribution – • Observations vary, so functions of the data also have variation The mean, x-bar, is most often used – • Examples: average, maximum, rank-sum 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: • N.B.: “two-sided” hypothesis H0 : X - W = 0 HA : X - W 0 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. -1 00 -3 -2 - 0 2 3 -80 -60 -40 -20 0 20 40 60 80 1 00 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 RR may 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. x=2.1 -1 00 -3 -2 - 0 2 3 -80 -60 -40 -20 0 20 40 60 80 1 00 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. x=-2.65 -1 00 -3 -2 - 0 2 3 -80 -60 -40 -20 0 20 40 60 80 1 00 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