Stat141 Oct 31 '06
2-group comparisons--further examples
Research design: Experiments vs non-experiments, cause and effect, what vs why?
Research design: Sample size calculations: how big n?

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> # Brogan-Kutner liver data--repeated measures analysis of variance

> bkdat = read.table(file="D:\\stat141\\brogkutrow.dat", header = T)
> # c4 method (1 = selective, 2=nonselective); c2 pre (urea synthesis); c3 post                   
> attach(bkdat)
> imp = post - pre
> tapply(imp,method,sort)
$"1"  [1] -6 -6 -5 -3 -3 -3 12 20
$"2"  [1] -24 -18 -18 -18 -18 -15 -14 -11  -8  -7  -6  -4   4

> t.test(imp~method)
        Welch Two Sample t-test
data:  imp by method 
t = 3.1743, df = 12.271, p-value = 0.007806
alternative hypothesis: true difference in means is not equal to 0 
95 percent confidence interval:
  4.044203 21.609643 
sample estimates:    mean in group 1 mean in group 2 
                            0.75000       -12.07692 
> # improvement diifers by methods--treatment by occasions interaction

> # t-test for equal var assumption equiv to repeated measures
> # in repeated measures analysis of variance 3.37^2 is F-statistic F(1,19)
> t.test(imp~method, var.equal = TRUE)
        Two Sample t-test
data:  imp by method 
t = 3.3709, df = 19, p-value = 0.003209
alternative hypothesis: true difference in means is not equal to 0 
95 percent confidence interval:
  4.862645 20.791201 
sample estimates:
mean in group 1 mean in group 2 
        0.75000       -12.07692 

                 pre,post description
> tapply(pre, method, summary)
$"1"
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  35.00   39.75   44.00   46.38   51.25   66.00 
$"2"
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  32.00   36.00   42.00   43.54   50.00   63.00 

> tapply(post, method, summary)
$"1"
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  35.00   41.25   47.00   47.13   54.25   60.00 
$"2"
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  14.00   18.00   32.00   31.46   36.00   67.00 

> # Duckworth BK improvement c = 10 + 2 = 12, p < .01

Further nonparametric exs, Kolmogorov-Smirnov test
comparing two empirical distributions or data against a specified standard
help(ks.test)   
ks.test                package:stats                R Documentation
Kolmogorov-Smirnov Tests
Description: Performs one or two sample Kolmogorov-Smirnov tests.

Exs: Verzani (SAT data example p. 269-270). PLOS cell random ex from 10/26

From R-docs do their examples:
> x = rnorm(50); y = runif(30) # both mean 0 symmetric dists
> ks.test(x,y) # Do x and y come from the same distribution?
        Two-sample Kolmogorov-Smirnov test
data:  x and y 
D = 0.6133, p-value = 4.25e-07
alternative hypothesis: two.sided 

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Research Design: Experiments vs Observational (non-experimental) Studies
Some exs in cause and effect determinations
  portacavel shunt (Gilbert, Light, Mosteller)
  prayer and surgery
  soda pop and cancer
  TV and ADHD (prior) TV violence and agression, reciprocal effects studies
Exs of experiments (from in the news): 
  Dodder vine, smoke alarms
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Research Design
Power and sample size, Power calculations via  power.t.test (see web page references)
Sample size calculations for Confidence Intervals easier (on overheads)

Large vs small effects SW p.274
mu1 = 69.1, mu2 = 64.1, => delta = 5, sigma = 2.5 => effect size = 2
> power.t.test(delta = 5, sd = 2.5, n=11) #find power given other parameters
     Two-sample t test power calculation 
              n = 11
          delta = 5
             sd = 2.5
      sig.level = 0.05
          power = 0.9937072
    alternative = two.sided
 NOTE: n is number in *each* group 

second ex p.274 mu2 = 63.5, mu2 = 64.1, power.t.test(delta = .6, sd = 2.5, n=11)
produces  power = 0.07727976  ; SW say <.10 p.274

SW Appendix 7.1  p. 623, how big n?
two sample t-test examples [find n for specied power--matches Table 5 p.678 entries]
      
>power.t.test(delta=.4,sd = 1, power = .8)  |> power.t.test(delta=.5,sd=1, power=.8)
     Two-sample t test power calculation    |     Two-sample t test power calculation   
              n = 99.08057                  |              n = 63.76576                 
          delta = 0.4                       |          delta = 0.5                      
             sd = 1                         |             sd = 1                        
      sig.level = 0.05  #SW did one-tail.025|      sig.level = 0.05                     
          power = 0.8                       |          power = 0.8                      
    alternative = two.sided                 |    alternative = two.sided                
 NOTE: n is number in *each* group          | NOTE: n is number in *each* group         

Also can do via 2 group case as one-way anova using power.anova.test (next week)
Also to come, power.prop.test Power calculations two sample test for proportions