LSU-HSC School of Public Health Biostatistics 1 Statistical Core Didactic Introduction to Biostatistics Donald E. Mercante, PhD

LSU-HSC School of Public Health Biostatistics 2 Randomized Experimental Designs

LSU-HSC School of Public Health Biostatistics 3 Randomized Experimental Designs Three Design Principles: 1. Replication 2. Randomization 3. Blocking

LSU-HSC School of Public Health Biostatistics 4 Randomized Experimental Designs 1. Replication Allows estimation of experimental error, against which, differences in trts are judged.

LSU-HSC School of Public Health Biostatistics 5 Randomized Experimental Designs Replication Allows estimation of expt’l error, against which, differences in trts are judged. Experimental Error: Measure of random variability. Inherent variability between subjects treated alike.

LSU-HSC School of Public Health Biostatistics 6 Randomized Experimental Designs If you don’t replicate You can’t estimate!

LSU-HSC School of Public Health Biostatistics 7 Randomized Experimental Designs To ensure the validity of our estimates of expt’l error and treatment effects we rely on...

LSU-HSC School of Public Health Biostatistics 8 Randomized Experimental Designs...Randomization

LSU-HSC School of Public Health Biostatistics 9 Randomized Experimental Designs 2. Randomization leads to unbiased estimates of treatment effects

LSU-HSC School of Public Health Biostatistics 10 Randomized Experimental Designs Randomization leads to unbiased estimates of treatment effects i.e., estimates free from systematic differences due to uncontrolled variables

LSU-HSC School of Public Health Biostatistics 11 Randomized Experimental Designs Without randomization, we may need to adjust analysis by stratifying covariate adjustment

LSU-HSC School of Public Health Biostatistics 12 Randomized Experimental Designs 3. Blocking Arranging subjects into similar groups to account for systematic differences

LSU-HSC School of Public Health Biostatistics 13 Randomized Experimental Designs Blocking Arranging subjects into similar groups (i.e., blocks) to account for systematic differences - e.g., clinic site, gender, or age.

LSU-HSC School of Public Health Biostatistics 14 Randomized Experimental Designs Blocking leads to increased sensitivity of statistical tests by reducing expt’l error.

LSU-HSC School of Public Health Biostatistics 15 Randomized Experimental Designs Blocking Result: More powerful statistical test

LSU-HSC School of Public Health Biostatistics 16 Randomized Experimental Designs Summary: Replication – allows us to estimate Expt’l Error Randomization – ensures unbiased estimates of treatment effects Blocking – increases power of statistical tests

LSU-HSC School of Public Health Biostatistics 17 Randomized Experimental Designs Three Aspects of Any Statistical Design Treatment Design Sampling Design Error Control Design

LSU-HSC School of Public Health Biostatistics 18 Randomized Experimental Designs 1. Treatment Design How many factors How many levels per factor Range of the levels Qualitative vs quantitative factors

LSU-HSC School of Public Health Biostatistics 19 Randomized Experimental Designs One Factor Design Examples –Comparison of multiple bonding agents –Comparison of dental implant techniques –Comparing various dose levels to achieve numbness

LSU-HSC School of Public Health Biostatistics 20 Randomized Experimental Designs Multi-Factor Design Examples: – Factorial or crossed effects Bonding agent and restorative compound Type of perio procedure and dose of antibiotic –Nested or hierarchical effects Surface disinfection procedures within clinic type

LSU-HSC School of Public Health Biostatistics 21 Randomized Experimental Designs 2. Sampling or Observation Design Is observational unit = experimental unit ? or, is there subsampling of EU ?

LSU-HSC School of Public Health Biostatistics 22 Randomized Experimental Designs Sampling or Observation Design For example, Is one measurement taken per mouth, or are multiple sites measured? Is one blood pressure reading obtained or are multiple blood pressure readings taken?

LSU-HSC School of Public Health Biostatistics 23 Randomized Experimental Designs 3. Error Control Design concerned with actual arrangement of the expt’l units How treatments are assigned to eu’s

LSU-HSC School of Public Health Biostatistics 24 Randomized Experimental Designs 3. Error Control Design Goal: Decrease experimental error

LSU-HSC School of Public Health Biostatistics 25 Randomized Experimental Designs 3. Error Control Design Examples: CRD – Completely Randomized Design RCB – Randomized Complete Block Design Split-mouth designs (whole & incomplete block) Cross-Over Design

LSU-HSC School of Public Health Biostatistics 26 Inferential Statistics Hypothesis Testing Confidence Intervals

LSU-HSC School of Public Health Biostatistics 27 Hypothesis Testing Start with a research question Translate this into a testable hypothesis

LSU-HSC School of Public Health Biostatistics 28 Hypothesis Testing Specifying hypotheses: H 0 : “null” or no effect hypothesis H 1 : “research” or “alternative” hypothesis Note: Only the null is tested.

LSU-HSC School of Public Health Biostatistics 29 Errors in Hypothesis Testing When testing hypotheses, the chance of making a mistake always exists. Two kinds of errors can be made: Type I Error Type II Error

LSU-HSC School of Public Health Biostatistics 30 Errors in Hypothesis Testing Reality   Decision H 0 TrueH 0 False Fail to Reject H 0  Type II (  ) Reject H 0 Type I (  ) 

LSU-HSC School of Public Health Biostatistics 31 Errors in Hypothesis Testing Type I Error –Rejecting a true null hypothesis Type II Error –Failing to reject a false null hypothesis

LSU-HSC School of Public Health Biostatistics 32 Errors in Hypothesis Testing Type I Error –Experimenter controls or explicitly sets this error rate -  Type II Error –We have no direct control over this error rate - 

LSU-HSC School of Public Health Biostatistics 33 Randomized Experimental Designs When constructing an hypothesis: Since you have direct control over Type I error rate, put what you think is likely to happen in the alternative. Then, you are more likely to reject H 0, since you know the risk level (  ).

LSU-HSC School of Public Health Biostatistics 34 Errors in Hypothesis Testing Goal of Hypothesis Testing –Simultaneously minimize chance of making either error

LSU-HSC School of Public Health Biostatistics 35 Errors in Hypothesis Testing Indirect Control of β Power –Ability to detect a false null hypothesis POWER = 1 - 

LSU-HSC School of Public Health Biostatistics 36 Steps in Hypothesis Testing General framework: Specify null & alternative hypotheses Specify test statistic and  -level State rejection rule (RR) Compute test statistic and compare to RR State conclusion

LSU-HSC School of Public Health Biostatistics 37 Steps in Hypothesis Testing test statistic Summary of sample evidence relevant to determining whether the null or the alternative hypothesis is more likely true.

LSU-HSC School of Public Health Biostatistics 38 Steps in Hypothesis Testing test statistic When testing hypotheses about means, test statistics usually take the form of a standardize difference between the sample and hypothesized means.

LSU-HSC School of Public Health Biostatistics 39 Steps in Hypothesis Testing test statistic For example, if our hypothesis is Test statistic might be:

LSU-HSC School of Public Health Biostatistics 40 Steps in Hypothesis Testing Rejection Rule (RR) : Rule to base an “Accept” or “Reject” null hypothesis decision. For example, Reject H 0 if |t| > 95 th percentile of t-distribution

LSU-HSC School of Public Health Biostatistics 41 Hypothesis Testing P-values Probability of obtaining a result (i.e., test statistic) at least as extreme as that observed, given the null is true.

LSU-HSC School of Public Health Biostatistics 42 Hypothesis Testing P-values  Probability of obtaining a result at least as extreme given the null is true.  P-values are probabilities  0 < p < 1 <-- valid range  Computed from distribution of the test statistic

LSU-HSC School of Public Health Biostatistics 43 Hypothesis Testing P-values  Generally, p<0.05 considered significant

LSU-HSC School of Public Health Biostatistics 44 Hypothesis Testing

LSU-HSC School of Public Health Biostatistics 45 Hypothesis Testing Example Suppose we wish to study the effect on blood pressure of an exercise regimen consisting of walking 30 minutes twice a day. Let the outcome of interest be resting systolic BP. Our research hypothesis is that following the exercise regimen will result in a reduction of systolic BP.

LSU-HSC School of Public Health Biostatistics 46 Hypothesis Testing Study Design #1: Take baseline SBP (before treatment) and at the end of the therapy period. Primary analysis variable = difference in SBP between the baseline and final measurements.

LSU-HSC School of Public Health Biostatistics 47 Hypothesis Testing Null Hypothesis: The mean change in SBP (pre – post) is equal to zero. Alternative Hypothesis: The mean change in SBP (pre – post) is different from zero.

LSU-HSC School of Public Health Biostatistics 48 Hypothesis Testing Test Statistic: The mean change in SBP (pre – post) divided by the standard error of the differences.

LSU-HSC School of Public Health Biostatistics 49 Hypothesis Testing Study Design #2: Randomly assign patients to control and experimental treatments. Take baseline SBP (before treatment) and at the end of the therapy period (post-treatment). Primary analysis variable = difference in SBP between the baseline and final measurements in each group.

LSU-HSC School of Public Health Biostatistics 50 Hypothesis Testing Null Hypothesis: The mean change in SBP (pre – post) is equal in both groups. Alternative Hypothesis: The mean change in SBP (pre – post) is different between the groups.

LSU-HSC School of Public Health Biostatistics 51 Hypothesis Testing Test Statistic: The difference in mean change in SBP (pre – post) between the two groups divided by the standard error of the differences.

Interval Estimation Statistics such as the sample mean, median, and variance are called point estimates -vary from sample to sample -do not incorporate precision

Interval Estimation Take as an example the sample mean: X ——————>  (pop n mean) Or the sample variance: S 2 ——————>  2 (pop n variance) Estimates

Interval Estimation Recall, a one-sample t-test on the population mean. The test statistic was This can be rewritten to yield:

Interval Estimation Confidence Interval for  : The basic form of most CI : Estimate ± Multiple of Std Error of the Estimate

Interval Estimation Example: Standing SBP Mean = 140.8, S.D. = 9.5, N = 12 95% CI for  : ± (9.5/sqrt(12)) ± (134.8, 146.8)