Analysis of Variance (ANOVA) for Percent Population Employed in Tarascan Villages, Slides of Statistics

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Parametric Analysis of Variance

The same principles used in the two-sample means test can be applied to more than two sample means.

In these cases we call the test analysis of variance.

The null hypothesis for analysis of variance (AOV or ANOVA) is:

H 0 : μ 1 = μ 2 … = μk

And this may be applied over both time and space.

Generic Data Set

Group1 Group2 Group

Obs 1 X1,1 X 2,1 X 3,

Obs 2 X1,2 X 2,2 X 3,

Obs 3 X1,3 X 2,3 X 3,

Obs 4 X1,4 X 2,4 X 3,

Obs 5 X1,5 X 2,5 X 3,

i

j

Note that i denotes rows and j denotes columns.

AOV:

• Compares the variation within the groups (columns) to the variation among the group means.

− If the variation among the group means is much greater than within the groups, reject H 0.

− If the variation among the group means is not much greater than within the groups, accept H 0.

( )

WSS TSS BSS

C n

X BSS

N

X TSS X C where C

i

i

i

n

i (^) i

n

j

ij

n

j

ij ij

k i

= −

−



 



=

= − =

∑

∑

∑

∑∑ ∑

=

=

= =

1

2

1

1

2 2 1

We will use this set of equations because they are easier to calculate.

We test for differences among the groups using the F test.

• The standard F test is simply a ratio of variances.
• Here the F test is slightly modified to take into account:
  1. The number of groups.
  2. Differences in the number of group members.

There are 2 types of degrees of freedom for the F statistic:

• k – 1 (where k is the number of columns) for the numerator.
• N – k (where N is the total sample size) for the denominator.

Note that the table from the book is different than the web table.

• Book p values are < or > a set alpha level.
• Web table p values can be a range.

A few words on Sum of Squares…

• In all cases, the sum of squares is a measure of total variation from the mean in a data set.
• If the sum of squares is large , then the data set has a lot of variation. This makes it more difficult to reject H (^) o (find a difference).
• If the sum of squares is small , then the data set has little variation. This makes it easier to reject H (^) o (find a difference).
• Once the sum of squares has been determined there are many ways of using the information.

Understanding the role of the total sum of squares (TSS) is straightforward: it is the total amount of variation in the data set.

2 = (^) ∑∑ ( − ) i j

TSS Xij Xij

( )

ni j

ij ij

k i N

TSS X C where C X 1

2 2 1

is equivalent to

For TSS the data are pooled together.

We are examining the distance between the means of each group (A and B).

The within sum of squares (WSS) can be thought of as the “error” variance.

• We want to factor out variation within the groups.

i j

WSS Xij X j

( )^2

WSS = TSS − BSS

is equivalent to

The F test is simply the ratio between the error variance (within groups) and the treatment variance (between groups).

IF the ratio is small then the difference is also small (accept H (^) o). IF the ratio is large then the difference is also large (reject H (^) o).

N k

WSS

k

BSS F

−

= −^1

Tarascan Village Percent Population Employed

Mountain Meseta Lake

We will drop the towns along the highways since we are interested only in rural towns.

Lake Group

Meseta Group

Mountain Group