# We worked with a stereogram fusion data set (described in the Powerpoint for today).

# Here are the data:

> fusion <- read.csv("http://faculty.ucmerced.edu/jvevea/classes/105/data/fusion.csv")
> head(fusion)
      Time Image
1 47.20001    No
2 21.99998    No
3 20.39999    No
4 19.70001    No
5 17.40000    No
6 14.70000    No
> attach(fusion)

# We compare means and standard deviations:

> tapply(Time, Image, mean)
      No      Yes 
8.560465 5.551429 
> tapply(Time, Image, sd)
      No      Yes 
8.085412 4.801739 

# Hmm... It looks like the two populations may differ both in their means
# and in their standard deviations, based on what we see in the samples.

# An additional complication is that both distributions are positively skewed:

> tapply(Time, Image, stem)

  The decimal point is 1 digit(s) to the right of the |

  0 | 2222222223333444
  0 | 5566678888999
  1 | 00002233
  1 | 57
  2 | 002
  2 | 
  3 | 
  3 | 
  4 | 
  4 | 7


  The decimal point is at the |

   0 | 01246778
   2 | 03478933568
   4 | 699
   6 | 00134
   8 | 6697
  10 | 
  12 | 
  14 | 49
  16 | 2
  18 | 7

$`   No`
NULL

$`   Yes`
NULL

> tapply(Time, Image, pskew)
       No       Yes 
0.6160965 1.2192014 
> 
> table(Image)
Image
    No    Yes 
    43     35 
> length(Time)
[1] 78
> qqnorm(Time[1:43]); qqline(Time[1:43])
> qqnorm(Time[44:78]); qqline(Time[44:78])

# Ignoring problems with independence, we proceed with a pooled-variance
# t test. The result is just barely nonsignificant:

> t.test(Time~Image, var.equal=TRUE)

        Two Sample t-test

data:  Time by Image
t = 1.9395, df = 76, p-value = 0.05615
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.0809383  6.0990099
sample estimates:
 mean in group    No mean in group    Yes 
            8.560465             5.551429 

# If we do the form of the t test that doesn't pool the variances and
# uses Satterthwaite's approximate degrees of freedom, the results are
# just barely significant:

> t.test(Time~Image)

        Welch Two Sample t-test

data:  Time by Image
t = 2.0384, df = 70.039, p-value = 0.04529
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 0.06493122 5.95314037
sample estimates:
 mean in group    No mean in group    Yes 
            8.560465             5.551429 


# In response to a student question, I showed you that you can always
# get the last n commands issued in R using the history(n) command:

> history(23)
> 

# We spent the rest of the class working with Bayesian analyses of the
# same data set, including two that essentially duplicate the t tests, and
# one that more appropriately models the data as gamma distributed rather
# than normally distributed. The code is posted.