MATH 251, P&S I, Fall 2007, Fri. Nov. 16, Day 36.after class.

_{What is the significance to Statistics of the Guinness Stout Bottle ?
~~~~~~~~~~~~~~~}
Homework questions:  Day 35

MATCHED PAIRS t procedures: (get for free!)   Example by hand, robustness  Day 35
SPSS:  Analyze >Compare Means> Paired-Samples T-test.  handout
        Data in parallel columns--subtracts rightmost from left column. Don't get to choose which way to subtract.
        CI level under Options.
 or  Transform>Compute:  Let Target variable be Difference, Numeric expression be  VarA -VarB.  You can use the Difference to examine for Normality, do one-sample procedures on Difference.

What if t's not suitable?
 Skewness:  Try log or other transformation, work on transformed data.  (Sadly, CI's can't be transformed back. Because  µ_log(X) is not equal to log(µ_X) ) last time.
 Outliers or other nonnormality:  Distribution-free/ nonparametric procedures.  Usually less power than distribution-based. (Uses less information, duh!)  Often based on binomial or similar models.

Sign test (p. 465-8) is a nice "trick", that turns any paired sample situation into a binomial situation.
For each pair, "success" is that the item from Group A is bigger than the matched item from Group B.  If there are ties, just throw them away (like the flipped coin that balances on its edge).
The null hypothesis is always that the groups are the same, so it is just like a coin-flip, the prob. of success is 1/2 under H₀. Then see how likely you are to get at least as many successes as you saw, using the binomial distribution.  That's the p-value, for the alternative  H_a that Group A is bigger on average than Group B.  More specifically, we're testing this:
H₀ : (the median of X_{Group A-GroupB}  is 0) ~ (probability that X_{Group A-GroupB} is positive = .5)  ~  ( p =.5) .
H_a:  (the median is above 0) ~ (probability that X_{Group A-GroupB} is positive > .5)  ~  ( p >.5) .

Example:  We suspect that students living on campus for their first semester gain weight.  Poll 11 students, asking just the sign of their weight change:
Get these results +  +  +  0  +  -  +  +  +  -  +  (0 means no change)   8 +'s and 2 -'s out of 10.
If there's no weight gain on average (Median gain is 0) we have a B(10, .5) distribution.  One sided alternative, that median gain is higher.  Let X be B(10, .5).  Then the P-value is P(X = 8, 9, or 10) = .0439 + .0098 + .0010 =  .0547, from Table C in the book.
SPSS:  Transform/Compute (pp.8&9, first handout)
    CDF.BINOM(7, 10, .5) gives the probability that X is less than or equal to 7, in a B(10, .5) distribution.  You will probably want to increase the number of digits after the decimal point (Decimals).  To find P (X = 8, 9, or 10), subtract the SPSS number from 1.

     Disadvantage:  You're obviously throwing away a lot of information (how big the differences are).  The result is that the power to detect a difference--if there is one--is much less than that of a t-test, where the t is usable.
     The sign test can be extended to a single data set, where you test the median:  If a is the median, then in the population, half the observations will be above a, and half below. Each data point is then like a coin flip, above or below the median.  (Can you see how this could be extended to test for a particular value of the first quartile, for instance?)
SPSS will do the sign test if you have the two "matched pair" variables.   (Be sure you have descriptive labels)
Analyze>Nonparametric Tests>2 Related Samples.  Get a box where you choose the pair (can't choose direction of subtraction).
Under Test Type, choose Sign.  Get counted results and two-sided P-value.
Start here Monday Yes
Sec. 7.2, Comparing two means"Two-sample tests".  Two SRS's, independent, from distinct  populations. (Populations are normally distributed)
Often--comparing means from an experiment with two treatments (usually control and "treatment"). Cf. p. 202.
                /--- Group 1, n₁---- Treatment 1---\
              /                                    \
 Random asst.                                       Compare results
              \                                    /
               \--- Group 2, n₂---- Treatment 2---/
To examine  the difference of the  two means, µ₁ - µ₂:
Theoretical assumption is normal populations.  Back to back stemplots are good; boxplots will do.
We use the Difference of the two x-bars,  diff =  xbar₁ - xbar₂ = .
  The Standard Deviation  is calculated like the hypotenuse of a right triangle (Pythagorean Theorem),  from the individual standard deviations:
 
 

Then the "Two-sample z-statistic"   is N(0,1) (p. 488)
But we don't know the population standard deviations!  We need the Standard Error of the difference  xbar₁ - xbar₂ , and then we can proceed as before, more or less. As usual, we substitute sample standard deviations for population standard deviations, and our z's are replaced by t's.

For testing, if H_o is "population means are equal"

"Two-sample t-statistic"

Unfortunately, this doesn't  have an exact t-distribution, and its exact distribution is very hard to deal with; but if we "adjust" the  degrees of freedom, t is a good approximation..

For doing by hand:  df = smaller of (n₁- 1) and (n₂- 1).
Will give a "conservative" result--slightly wider C.I., slightly less significance, than a "sharper" value.  If your results hinge on the difference between this result and the computer result, they're too close for comfort anyway. Table D? go to lower df. if the one you want isn't given.

From a computer:  df = complicated formula on p. 498.  Produces non-integer degrees of freedom.  Very good approximation to the exact distribution, if both sample sizes are at least 5. Unsuitable for doing by hand.

Once we have (xbar₁ - xbar₂) , SE_diff ,  and the df, our formulas pattern on the earlier ones.
CI :  estimate + t* ^. SE_estimate
    CI for µ₁ - µ₂, difference of means,  is 
Test:  H₀: µ₁ - µ₂ = 0 same as µ₁ = µ₂ , "no difference" always
        H_a: µ₁ - µ₂ > 0 same as µ₁ > µ₂ Be careful with these, that you know which direction you want.
    or H_a: µ₁ - µ₂ < 0 same as µ₁ < µ₂ Often we label our variables "1" and "2" so that we expect µ₁ > µ₂
    or H_a: µ₁ - µ₂ 0 same as µ₁  µ₂  (not equal)
        Calculate t, find P-value (approximate, conservative) Example by hand. 
 You can check your by-hand work with  Excel Two-sample calculator

Robust? Yes...p. 493  Outliers are bad, as before: Use same guidelines (p. 463) with n = n₁ + n₂
      Large n's have robustness from CLTh.
   Equal sample sizes help: then robust against non-normality , more so if populations have the same shape, down to n=5 each..
             In doubt? Use the conservative df!

--SPSS will do our computations when we are given raw data. Next.