Statistics Review – Part II Topics: – Hypothesis Testing – Paired Tests – Tests of variability 1

2 Sampling and the Normal Distribution inference population sample Now that we have a good sampling plan, how do we monitor the population?

3 Sampling and the Normal Distribution

Suppose the standard of care for diabetics in the ICU is to have HbA1c level  0 = 9.5 You want to see whether your ICU diabetic patients in the ICU meet the standard of care What do you do next? 4 Tests of Means

Sample your diabetic patients in the ICU Randomization…. Sample all diabetic patients every day? – Independence between measurements Time of day? – Controlling for external factors 5 Tests of Means

Randomization – Control for all known factors – Then, randomize the rest – Uncontrollable factors will randomly affect your diabetic patients’ measurements Uncontrollable factors will ‘average out’ 6 Tests of Means

Develop a sampling plan…. What are the controllable elements? How do we randomize and collect our blood sugar measurements? 7 Tests of Means

Suppose we collect A1c data on 20 diabetic patients in the ICU – Mean 7.5 – Standard deviation 4.0 How do we proceed? 8 Tests of Means

Identify the null and alternative hypothesis – What is it we’re testing? Equal versus greater than (left-tailed test) – Ho: mean A1c = 9.5 – H1: mean A1c < Tests of Means

10 Compute the test statistic which follows Student’s t-distribution with n – 1 degrees of freedom. Tests of Means

Our hypothesis test expressed on the normal distribution Tests of Means

Calculate the t-statistic – 19 degrees of freedom T = = – sqrt(20) Tests of Means

Compare to the critical value of the t-statistic with 19 degrees of freedom – What do you conclude? Do your conclusions match you graphical interpretation? 13 Tests of Means

Instead, say that you sampled 80 diabetic patients in the ICU Mean = 7.6 Standard deviation = 4.0 What do you conclude now? 14 Tests of Means

T = = 2.46 How much area is in the ‘curve’ now? What are we basing our assumptions on? – How do we validate our assumptions? – sqrt(80) Tests of Means

16 A P-value is the probability of observing a sample statistic as extreme or more extreme than the one observed under the assumption the null hypothesis is true. Tests of Means

P-value of 2.46 is of 100 experiments will have a t value greater than 2.46 due to ‘random chance’ 1% chance that 2.46 occurred due to chance – more likely that there is a difference between the mean A1c values If p-value < 0.05, we often say we reject the null hypothesis in favor of the alternative (there is strong evidence to reject the null hypothesis). 17 Tests of Means

18 Lastly, we have to validate the assumptions under which we based our test. Here, we use a normal probability plot. Fat pencil test to detect normality Tests of Means

Example: – Is the turn-around time in the lab less than 120 minutes? – Null Hypothesis: – Alternative Hypothesis: 19 Tests of Means

20 Is the turn-around time in the lab less than 120 minutes? 100 samples Mean = Standard deviation = 25.8 t=6.98 df=99 p-value Tests of Means

What is the ‘strength’ of the evidence against the null hypothesis? What do you conclude relative to the null hypothesis? 21 Tests of Means

End: Z-statistics, T-statistics Start: Independent samples t-test 22 Tests of Means

Objective: – Determine whether there are statistically significant differences between means of independently-drawn samples Example: – Mean LOS is the same for 2 physicians – Mean registration time is the same on Fridays as it is on Tuesdays 23 Tests of Means

24 Tests of Means

25 Tests of Means

Claim: – Physician A avg LOS = Physician B avg LOS Alternative Hypothesis: – Physician A avg LOS not equal to Physician B avg LOS 26 Tests of Means

27 Compare the average LOS for visual differences: Tests of Means

Graphically, we have a distribution for the distribution of the difference of avg LOS 28 Tests of Means

Hypothesis: Physician A and B have the same average LOS – Sample 12 Physician A and B patients – A: Mean LOS is 5.8 Standard deviation is 4.0 – B: Mean LOS is 8.5 Standard deviation is Tests of Means

Calculate the test statistic 30 Tests of Means

31 The degrees of freedom used to determine the critical value(s) presented in the last example are conservative. Results that are more accurate can be obtained by using the following degrees of freedom: Tests of Means

What is the t-statistic? What is the p-value associated with the two-sided test? What do you conclude? 32 Tests of Means

Say, now that you sample 25 patients from each physician: – Physician A and B patients: – A: Mean LOS is 6.8 Standard deviation is 4.3 – B: Mean LOS is 8.1 Standard deviation is Tests of Means

What is the t-statistic? What is the p-value associated with the two-sided test? What do you conclude? 34 Tests of Means

35 Lower Bound = Upper Bound = Tests of Means

End: Independent Samples Test Start: Tests of Variability – F-Tests 36 Tests of Means

When might you want to know whether the variability in 2 populations is the same? – Time series - whether a process is ‘in control’; or – Comparison between two populations 37 Tests of Variances

38 Tests of Variances

39 Tests of Variances

40 Comparing variability is based on the ratio of sample variances The ratio of sample variances is called the F-statistic and follows the F-distribution Tests of Variances

Since, If is large relative to, then F is very large and suggests the sample variances are different. If is large relative to, then F is very small and suggests the sample variances are different. 41 Tests of Variances

How ‘large’ or how ‘small’ does F have to be to be ‘significant’? – That depends on the F-distribution which takes into account the sample sizes for both sample variances – Recall, the larger the sample size, the more precisely we can characterize the sample variance (and test for differences between the sample variances) 42 Tests of Variances

43 Tests of Variances

44 Testing Claims Regarding Two Population Standard Deviations 1.The samples are independent simple random samples from both populations. Drawing observations in one population does not affect the drawing of observations in the second population 2.The populations from which the samples are drawn are normally distributed. If the populations from which the samples are drawn are not normally distributed, do not use the F-test for equality of variances Tests of Variances

45 Tests of Variances

46 Tests of Variances

47 Testing hypotheses about equality of variances: Is the critical F with n 1 – 1 degrees of freedom in the numerator and n 2 – 1 degrees of freedom in the denominator and an area of α to the right of the critical F. Tests of Variances

48 Tests of Variances

49 Tests of Variances

50 Tests of Variances

Example: – Change process in lab. 51 Tests of Variances

Summary: – Pre-change Minimum: 51Maximum: 236 Mean: 128 minutes Standard deviation: 40 minutes – Post-change Minimum: 105Maximum: 143 Mean: 121 minutes Standard deviation: 10 minutes 52 Tests of Variances

50 weeks pre- and post- Calculate the F-test F = 40/10 = 4 Look it up! P-value! – Less than critical value? 53 Tests of Variances

What do you conclude about the equality of the variability of the distributions? Do this ‘intuitively’ makes sense when compared to the graph? 54 Tests of Variances

Conclusions Review: – Paired t-tests – Independent samples t-test – F-test for equality of variances – Hypothesis testing – P-values – Confidence intervals 55