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How the Results are Changed When The Independent Variable Is Effective


Visualizing the Effect of the Independent Variable

Expected Results when the Independent Variable is NOT effective

These are the kind of results we expect when the Independent Variable is NOT effective

  • The range of scores for each level of the independent variable is about the same.
  • The sample means differ by sampling error; that is, they differ by random flucuation.
  • The entire range of scores therefore is about the same as the range of any single group of subjects.

Expected Results when the Independent Variable IS Effective

These are the kind of results we expect when the Independent Variable is Effective

  • The range of scores for each level of the independent variable is about the same.
  • The sample means are offset from one another by more than sampling error
  • The entire range of scores is therefore wider than the range of any single group.

Quantifying "Was the Independent Variable Effective?"

Remember
Variance is Standard Deviation Squared
Another name for Variance is "Mean Square"
Variance is "Sum of Squares" divided by "Degrees of Freedom"
"Mean Square" is the "average Sum of Squares"

The Rationale

Source Table for Independent Groups ANOVA

Source

DF
SS
MS
F
Numerator of F

Between Groups

2

90013.3423

45006.6712

4.5235

Denominator of F

Within Groups
AKA
"Error Variance"

12

119393.6000

9949.4667

Total

14

209406.9423

Denominator of the F-Ratio. This is always an estimate of population variance

Error Variance

An estimate of population variable that does not depend upon whether or not the indpendent variable was effective.

  • How much subjects differ within a group isn't changed by whether or not the independent variable is effective.
    • This variability is intrinsic variation; the degree to which subjects respond differently when they are treated exactly the same (as are the subjects within one experimental condition).
    • Intrinsic variation is the population variability --
    • We can estimate population variance by calculating the variance of each group of subjects.
      • If we had three groups in the experiment, we'd have three estimates of population variance. The variance is averaged in a special way (called "pooling") to provide one measure of intrinsic variation. That measure is called ERROR VARIANCE.
Within Groups SS
Calculate the Sum of Squares for each group of subjects. The degrees of freedom for each of these sums is one less than the number of scores in that group. Add together the sum of squares for the groups to get the Within Groups Sum of Squares.
Error Variance
aka
Mean Square Within Groups
Divide "Within Groups SS" by the DF
Numerator of the F-ratio. Its meaning depends upon whether or not the Ho is True or False.

Between Groups Variance

If the independent variable was NOT effective then this number estimates population variance

If the independent variable WAS effective, then this number is larger than population variance.

  • Use the group means and calculate the variance of the group means.
    • If the independent variable is NOT effective, then:
      • The group means differ from one another only because of sampling error.
      • This number estimates the population variance error of the means -- -- the [sem squared].
      • Multiply the estimate of variance error by n to estimate population variance. This number is known as the between groups variance.
        • Bet. Gps. Var. = pop. var.
      • The "Bet Gps Var" is an estimate of population variance and therefore is about the same value as the
    • If the independent variable IS effective then:
      • The group means differ from one another by sampling error AND by the operation of the independent variable--it has offset the means from one another.
        • Bet. Gps. Var. = pop. var. + effect of I.V.
        • The "effect of I.V." is a squared number; therefore it is always positive.
      • Between Groups Variance is larger than population variance.
Between Groups SS For its calculation, see below.
Between Groups Mean Square Divide the Between Groups SS by (the number of groups less 1).

Quantitative Effect of the Independent Variable

For each score

Bet Gps SS = Total SS - Within Gps SS

Calculate the "Average" Sum of Squares (aka "Mean Square")


What decision about the independent variable do you make here?

Here is a less obvious situation. Was the independent variable effective?


ANOVA Source Tables for the data displayed in the three graphics

The Independent Variable is NOT Effective

Source Table for One-Way Independent Groups ANOVA

Source

DF
SS
MS
F
p

Between Groups

2

33157.2022

16578.6011

1.2471

Within Groups

12

159518

13293.20000

Total

14

192675.6022


The Independent Variable is Effective

Source Table for One-Way Independent Groups ANOVA

Source

DF
SS
MS
F
p

Between Groups

2

90013.3423

180545.6712

4.5235

Within Groups

12

119393.6000

9949.4667

Total

14

209406.9423


Was the Independent Variable Effective?

Source Table for One-Way Independent Groups ANOVA

Source

DF
SS
MS
F
p

Between Groups

2

68824.9385

34412.4692

2.5486

Within Groups

12

162028.400

13502.3667

Total

14

230853.3385


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