How IV Affects Results

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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

The test statistic is "F"
F is a variance ratio
The denominator of F estimates the intrinsic variability when subjects are treated exactly the same
- The denominator is an estimate of population standard deviation squared
The numerator of F measures the variability among the group means
- If the independent variable is NOT effective, the numerator is an estimate of population variance
- if the indpendent variable IS effect, the numberator is an estimate of population variance plus the offset caused by the I.V.

The value of F is used to decide whether or not the independent variable is effective.
- F = 1 (approximately) Then the I.V. is NOT effective.
  - The numerator estimates population variance
  - The denominator estimates population variance.
- F > 1 (by a fair amount). Then the I.V. IS effective.
  - The numerator estimates population variance plus the offset caused by the independent variable
  - The denominator estimates population variance

	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

Total Sum of Squares
- How far is each score from the average of all the scores (deviation from the Overall Mean)
- Square each deviation and then sum all the deviations
Within Groups Sum of Squares
- How far is each score from the average of its own group (deviation from the Group Mean)
- Square each deviation and then sum all the deviations

Bet Gps SS = Total SS - Within Gps SS

Between Groups Sum of Square
- Can be calculated by subtraction.
- The variation left over must be because the group means are offset.
Replace each score with the sample average of its group.
- How far is each replaced score (a group mean) from the overall mean (deviation from the Overall Mean).
- Square each of these deviations and sum all the deviations.
- The Sum of Squares only has variation of the group means.

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

Between Groups Mean Square
- Between Groups SS divided by Degrees of Freedom
Within Groups Mean Square
- Within Groups SS divided by Degrees of Freedom

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