and One-Factor[Jain, Ch. 20]
and One-Factor
.
Number of bytes required by five programmers to code workload on three processors (R, V, and Z):
Here, r=5 and a=3.
Columns correspond to levels; rows correspond to replicas.
design! That's because i is a row, and j is a
column.
and 
are two notations for the same quantity!
), averaged over
all replicas i
in design
) and effect of factor j (i.e., )
where:
Equation used for prediction:
Recall our model:
Parameter values are:
Thus effect due to level j (i.e., ) is difference
between mean of observations at level j and grand mean.
Form column sums to derive and effects :
From table:
Recall our equation for predicted response:
Therefore:
Recall definition of :
Difference between (1) observed response for replica i of factor level j and (2) sum of the mean response and effect of factor j
Formally,
Recall:
,
, and
.
Thus:
Recall from linear regression our analysis to explain how much variation is due to
explained variation (i.e., the factor)
versus
unexplained variation (i.e., the experimental error).
The two quantities were SSR/SST and SSE/SST, respectively.
For one-factor design, we compute SSA/SST and SSE/SST.
So what are SSA, SSE, and SST?
Recall our model:
Squaring both sides, and adding equations for all i and j yields:
Thus:
Simplifying:
First note the following relationship:
But by definition SST = SSA + SSE. Therefore:
Recall that the observations are:
and that the formulas are:
where
Thus:
percent of variation explained by processors is
The remaining 89.6% is due to experimental error!
Allocation of variance shows that 89.6% of variation is due to error.
Why is this so high?
How do we identify which case is true?
We need a test to compare SSA and SSE.
How about using their ratio:
This ratio is not quite right - if we have many more replicas than
factor levels, SSE will be larger! So use ``SSE per sample'' and
``SSA per sample''. Let 
and 
denote, respectively,
the degrees of freedom of SSA and SSE:
How do factor in the sample ``quality'' (size)?
We need a statistical test to compare SSA and SSE.
Distribution:
. The parameter is the Greek letter ``nu,'' and
denotes the number of unit normals added together.
, variance is 
.
values.
variates:
distributions in the numerator and denominator.
values.
Tests following null hypothesis:
Response variable does not depend on any effect
Acceptance criteria:
Ratio does not exceed-quantile of the F distribution.
Note: As 
, -quantile
. So all F values in the tables exceed one. This is
intuitive:
So the question
Is factor statistically significant?
is equivalent to rejecting null hypothesis, or asking:
Does F statistic computed from our data exceed theoretical F?
Test gives ``yes/no'' answer for a chosen significance level to question of whether contribution of factor to variation is statistically significant.
ANOVA is a statistical procedure to compare the contribution of the percentages of variation attributed to the factor and the error.
Computed F statistic is 
:
If
The theoretical F is 
.
In the code size comparison:
Thus
The value of F[.90; 2,12] = 2.8. Because 0.7 ;SPMlt; 2.8, the test yields ``no significance.''
<Insert Tables 20.3 and 20.4 here!>
The one-factor analysis requires the same assumptions as were
used earlier for the design. Consider two of these:
We can visually test (1) by a quantile-quantile plot:
<Insert Fig. 14.9!>
Visually test (2) by plotting residuals versus predicted response:
<Insert Figs. 14.7, 14.8!>
Visually test (3) by last plot, but look for increasing spread:
<Insert Fig. 14.10!>
<Insert Fig. 18.2!>
<Insert Fig. 18.1!>
The estimated response is:
All three quantities are random variables because they are based on one sample (set of experiments).
As discussed earlier for designs, confidence interval for a
term x (for 
) in above equation is
the mean plus or minus the product of a t distribution value and
the standard deviation of x.
Note: Degrees of freedom for constructing CI's always equals DF for errors (tabulated in ANOVA table). It's a(r-1) because the errors for all replicas at a given factor level sum to one, so only r-1 values are unique for each of a factor levels. Jain incorrectly writes the formula as a( r-1) on page 335.
The values of 
are:
where standard deviation of errors is
Example: Consider again the code size comparison:
For 90% confidence...
None of the processor effect ( ) confidence intervals contains
zero.
Thus we cannot say with 90% confidence that the processors have a significant effect on code size.
Question: Is the code size required for processor R significantly larger or smaller than the code size for V?
Answer: Does confidence interval for contrast formulaexclude zero?
We will find that 90% confidence interval is:
Interval includes zero; therefore, we cannot state confidently that R requires more or less storage than V.
Note: Jain contains error on p. 337. He incorrectly lists interval as (-88.7, 111.1).
To compute , we use a contrast formula:
where 
and 
and 
.
From Table 20.3 in Jain:
Thus standard deviation of 
(from table
above, not ``56.1'' as Jain writes on p. 337).
Also need mean difference in effects:
Thus 90% confidence interval is
