Ch 24 - One-way ANOVA: Comparing several means Flashcards
When comparing >2 populations, the question is not only ____ but also____
whether each population mean µi is different from the others,
whether they are significantly different when taken as a group.
The first step in examining multiple populations statistically is
If that ____ showed statistical significance, _______
to test for an overall statistical significance as evidence of any difference among the parameters we want to compare. (ANOVA F test)
overall test
then a detailed follow-up analysis can examine all pair-wise parameter comparisons to define which parameters differ from which and by how much. (more complex methods (see Chapter 26))
Variance and standard deviation
If we have sampled several times from the same population with mean μ and standard deviation σ, then
If we have sampled from distinct populations with the same standard deviation σ but different means μ, then
A factor is
a variable that can take one of several levels used to differentiate one group from another.
An experiment has a one-way or ____ design if ____
completely randomized (one-way designs)
several levels of one factor are being studied and the individuals are randomly assigned to its levels. (There is only one way to group the data.)
One way vs two way ANOVA
The analysis of variance F test compares
the variation due to specific sources (levels of the factor) with the variation among individuals who should be similar (individuals in the same sample).
The analysis of variance F statistic for comparing several means is
F large vs small
ANOVA assumptions
Equal sample sizes and ANOVA
Equal sample sizes make the ANOVA more robust to deviations from the equal σ rule.
A simple and conservative approach: The ANOVA F test is approximately correct when
the largest sample variance is no more than ~ 4 times as large as the smallest sample variance.
There are tests to check for equality of variance. However, they tend to be sensitive to
deviations from the Normality assumption or require equal sample sizes
We have k
independent SRSs, from k populations or treatments.
The i^th population has a
Normal distribution with unknown mean µi.
All k populations have
the same standard deviation σ, unknown.
Under H0, all k samples
come from the same population N(μ,σ) and sample averages should be no more variable than points in individual samples.
When H0 is true, F
has the F distribution with k − 1 (numerator)
and N − k (denominator) degrees of freedom
MSG
the mean square for groups,
is a variance of the means weighted for sample size.
It measures the variability of the sample averages.
MSE
the mean square for error or pooled sample variance sp2
is the average sample variance weighted for sample sizes.
It measures the variability within each of the groups.
The F distribution is
asymmetrical and
has two distinct degrees of freedom.
This was discovered by Fisher, hence the label “F”.
Critical values on F chart
A two-sample t test vs ANOVA
A two-sample t test - assuming equal variance with a two-sided Ha and
ANOVA - comparing only 2 groups will give the same P-value.
The t test is more ____ (vs ANOVA)
more flexible
You may choose a one-sided alternative or you may want to run a t test assuming unequal variance if you are not sure that the 2 populations have the same standard deviation s.
Interpreting results from an ANOVA (H0 vs Ha)
H0 states that all ui are equal
Ha states that H0 is not true.
A significant P-value leads you to reject H0 indicating that not all ui are equal.
MSE
the mean square for error or pooled sample variance sp2,
estimates the common variance σ 2 of the k populations.
Thus, we can easily calculate separate level C confidence intervals for each population mean µI (often provided by software packages):
