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ANOVA

- ANOVA is used to compare means.
- However, if a difference is detected, and more than two means are being compared, ANOVA cannot tell you where the difference lies.
- In order to figure out which means differ, you can do a series of tests:
- Planned or unplanned comparisons of means.

PLANNED or A PRIORI CONTRASTS

- A comparison between means identified as being of utmost interest during the design of a study, prior to data collection.
- You can only do one or a very small number of planned comparisons, otherwise you risk inflating the Type 1 error rate.
- You do not need to perform an ANOVA first.

UNPLANNED or A POSTERIORI CONTRASTS

- A form of “data dredging” or “data snooping”, where you may perform comparisons between all potential pairs of means in order to figure out where the difference(s) lie.
- No prior justification for comparisons.
- Increased risk of committing a Type 1 error.
- The probability of making at least one type 1 error is not greater than α= 0.05.

PLANNED ORTHOGONAL AND NON-ORTHOGONAL CONTRASTS

- Planned comparisons may be orthogonal or non-orthogonal.
- Orthogonal: mutually non-redundant and uncorrelated contrasts (i.e.: independent).
- Non-Orthogonal: Not independent.
- For example:
- 4 means: Y1 ,Y2, Y3, and Y4
- Orthogonal: Y1- Y2 and Y3- Y4
- Non-Orthogonal: Y1-Y2 and Y2-Y3

ORTHOGONAL CONTRASTS

- Limited number of contrasts can be made, simultaneously.
- Any set of contrasts may have k-1 number of contrasts.

ORTHOGONAL CONTRASTS

- For Example:
- k=4 means.
- Therefore, you can make 3 (i.e.: 4-1) orthogonal contrasts at once.

HOW DO YOU KNOW IF A SET OF CONTRASTS IS ORTHOGONAL?

- ∑cijci’j=0
- where the c’s are the particular coefficients associated with each of the means and the i indicates the particular comparison to which you are referring.
- Multiply all of the coefficients for each particular mean together across all comparisons.
- Then add them up!
- If that sum is equal to zero, then the comparisons that you have in your set may be considered orthogonal.

HOW DO YOU KNOW IF A SET OF CONTRASTS IS ORTHOGONAL?

- For Example:

(After Kirk 1982)

- (c1)Y1 + (c2)Y2 = Y1 – Y2
- Therefore c1 = 1 and c2 = -1 because
- (1)Y1 + (-1)Y2 = Y1-Y2

ORTHOGONAL CONTRASTS

- There are always k-1 non-redundant questions that can be answered.
- An experimenter may not be interested in asking all of said questions, however.

PLANNED COMPARISONS USING A t STATISTIC

- A planned comparison addresses the null hypothesis that all of your comparisons between means will be equal to zero.
- Ho=Y1-Y2=0
- Ho= Y3-Y4=0
- Ho= (Y1+Y2)/2 –(Y3+Y4)/2
- These types of hypotheses can be tested using a t statistic.

PLANNED COMPARISONS USING A t STATISTIC

- Very similar to a two sample t-test, but the standard error is calculated differently.
- Specifically, planned comparisons use the pooled sample variance (MSerror)based on all k groups (and the corresponding error degrees of freedom) rather than that based only on the two groups being compared.
- This step increases precision and power.

PLANNED COMPARISONS USING A t STATISTIC

- Evaluate just like any other t-test.
- Look up the critical value for t in the same table.
- If the absolute value of your calculated t statistic exceeds the critical value, the null hypothesis is rejected.

PLANNED COMPARISONS USING A t STATISTIC: NOTE

- All of the t statistic calculations for all of the comparisons in a particular set will use the same MSerror.
- Thus, the tests themselves are not statistically independent, even though the comparisons that you are making are.
- However, it has been shown that, if you have a sufficiently large number of degrees of freedom (40+), this shouldn’t matter.

(Norton and Bulgren, as cited by Kirk, 1982)

PLANNED COMPARISONS USING AN F STATISTIC

- You can also use an F statistic for these tests, because t2 = F.
- Different books prefer different methods.
- The book I liked most used the t statistic, so that’s what I’m going to use throughout.
- SAS uses F, however.

CONFIDENCE INTERVALS FOR ORTHOGONAL CONTRASTS

- A confidence interval is a way of expressing the precision of an estimate of a parameter.
- Here, the parameter that we are estimating is the value of the particular contrast that we are making.
- So, the actual value of the comparison (ψ) should be somewhere between the two extremes of the confidence interval.

CONFIDENCE INTERVALS FOR ORTHOGONAL CONTRASTS

- The values at the extremes are the 95% confidence limits.
- With them, you can say that you are 95% confident that the true value of the comparison lies between those two values.
- If the confidence interval does not include zero, then you can conclude that the null hypothesis can be rejected.

ADVANTAGES OF USING CONFIDENCE INTERVALS

- When the data are presented this way, it is possible for the experimenter to consider all possible null hypotheses – not just the one that states that the comparison in question will equal 0.
- If any hypothesized value lies outside ofthe 95% confidence interval, it can be rejected.

CHOOSING A METHOD

- Orthogonal tests can be done in either way.
- Both methods make the same assumptions and are equally significant.

ASSUMPTIONS

- Assumptions:
- The populations are approximately normally distributed.
- Their variances are homogenous.
- The t statistic is relatively robust to violations of these assumptions when the number of observations for each sample are equal.
- However, when the sample sizes are not equal, the t statistic is not robust to the heterogeneity of variances.

HOW TO DEAL WITH VIOLATIONS OF ASSUMPTIONS

- When population variances are unequal, you can replace the pooled estimator of variance, MSerror, with individualvariance estimators for the means that you are comparing.
- There are a number of possible procedures that can be used when the variance between populations is heterogeneous:
- Cochran and Cox
- Welch
- Dixon, Massey, Satterthwaite and Smith

TYPE I ERRORS AND ORTHOGONAL CONTRASTS

- For C independent contrasts at some level of significance (α), the probability of making one or more Type 1 errors is equal to:
- 1-(1-α)C
- As the number of independent tests increases, so does the probability of committing a Type 1 error.
- This problem can be reduced (but not eliminated) by restricting the use of multiple t-tests to a priori orthogonal contrasts.

A PRIORI NON-ORTHOGONAL CONTRASTS

- Contrasts of interest that ARE NOT independent.
- In order to reduce the probability of making a Type 1 error, the significance level (α) is set for the whole family of comparisons that is being made, as opposed to for each individual comparison.
- For Example:
- Entire value of α for all comparisons combined is 0.05.
- The value for each individual comparison would thus be less than that.

WHEN DO YOU DO THESE?

- When contrasts are planned in advance.
- They are relatively few in number.
- BUT the comparisons are non-orthogonal (they are not independent).
- i.e.: When one wants to contrast a control group mean with experimental group means.

DUNN’S MULTIPLE COMPARISON PROCEDURE

- A.K.A.: Bonferoni t procedure.
- Involves splitting up the value of α among a set of planned contrasts in an additive way.
- For example:
- Total α = 0.05, for all contrasts.
- One is doing 2 contrasts.
- α for each contrast could be 0.025, if we wanted to divide up the α equally.

DUNN’S MULTIPLE COMPARISON PROCEDURE

- If the consequences of making a Type 1 error are not equally serious for all contrasts, then you may choose to divide α unequally across all of the possible comparisons in order to reflect that concern.

DUNN’S MULTIPLE COMPARISON PROCEDURE

- This procedure also involves the calculation of a t statistic (tD).
- The calculation involved in finding tD is identical to that for determining t for orthogonal tests:

DUNN’S MULTIPLE COMPARISON PROCEDURE

- However, you use a different table in order to look up the critical value (tDα;C,v).
- Your total α value (not the value per comparison).
- Number of comparisons (C).
- And v, the number of degrees of freedom.

DUNN’S MULTIPLE COMPARISON PROCEDURE: ONE-TAILED TESTS

- The table also only shows the critical values for two-tailed tests.
- However, you can determine the approximate value of tDα;C,v for a one-tailed test by using the following equation:
- tDα;C,v≈ zα/C + (z3α/C + zα/C)/4(v-2)
- Where the value of zα/C can be looked up in yet another table (“Areas under the Standard Normal Distribution”).

DUNN’S MULTIPLE COMPARISON PROCEDURE

- Instead of calculating tD for all contrasts of interest, you can simply calculate the critical difference (ψD) that a particular comparison must exceed in order to be significant:
- ψD = tDα/2;C,v √(2MSerror/n).
- Then compare this critical difference value to the absolute values of the differences between the means that you compared.
- If they exceed ψD, they are significant.

DUNN’S MULTIPLE COMPARISON PROCEDURE

- For Example:
- You have 5 means (Y1 through Y5).
- ΨD = 8.45
- Differences between means are:

- Those differences that exceed the calculated value of ΨD (8.45, in this case) are significant.

(After Kirk 1982)

DUNN-SIDAK PROCEDURE

- A modification of the Dunn procedure.
- t statistic (tDS) and critical difference (ψDS).
- There isn’t much difference between the two procedures at α < 0.01.
- However, at increased values of α, this procedure is considered to be more powerful and more precise.
- Calculations are the same for t and ψD.
- Table is different.

DUNN-SIDAK PROCEDURE

- However, it is not easy to allocate the total value of α unevenly across a particular set of comparisons.
- This is because the values of α for each individual comparison are related multiplicatively, as opposed to additively.
- Thus, you can’t simply add the α’s for each comparison together to get the total value of α for all contrasts combined.

DUNNETT’S TEST

- For contrasts involving a control mean.
- Also uses a t statistic (tD’) and critical difference (ψD’).
- Calculations are the same for t and ψ.
- Different table.
- Instead of C, you use k, the number of means (including the control mean).
- Note: unlike Dunn’s and Dunn-Sidak’s, Dunnet’s procedure is limited to k-1 non-orthogonal comparisons.

CHOOSING A PROCEDURE : A PRIORI NON-ORTHOGONAL TESTS

- Often, the use of more than one procedure will appear to be appropriate.
- In such cases, compute the critical difference (ψ) necessary to reject the null hypothesis for all of the possible procedures.
- Use the one that gives the smallest critical difference (ψ) value .

A PRIORI ORTHOGONAL and NON-ORTHOGONAL CONTRASTS

- The advantage of being able to make all planned contrasts, not just those that are orthogonal, is gained at the expense of an increase in the probability of making Type 2 errors.

A PRIORI and A POSTERIORI NON-ORTHOGONAL CONTRASTS

- When you have a large number of means, but only comparatively very few contrasts, a priori non-orthogonal contrasts are better suited.
- However, if you have relatively few means and a larger number of contrasts, you may want to consider doing an a posterioritest instead.

A POSTERIORI CONTRASTS

- There are many kinds, all offering different degrees of protection from Type 1 and Type 2 errors:
- Least Significant Difference (LSD) Test
- Tukey’s Honestly Significant Difference (HSD) Test
- Spjtotvoll and Stoline HSD Test
- Tukey-Kramer HSD Test
- Scheffé’s S Test
- Brown-Forsythe BF Procedure
- Newman-Keuls Test
- Duncan’s New Multiple Range Test

A POSTERIORI CONTRASTS

- Most are good for doing all possible pair-wise comparisons between means.
- One (Scheffé’s method) allows you to evaluate all possible contrasts between means, whether they are pair-wise or not.

CHOOSING AN APPROPRIATE TEST PROCEDURE

- Trade-off between power and the probability of making Type 1 errors.
- When a test is conservative, it is less likely that you will make a Type 1 error.
- But it also would lack power, inflating the Type 2 error rate.
- You will want to control the Type 1 error rate without loosing too much power.
- Otherwise, you might reject differences between means that are actually significant.

DOING PLANNED CONTRASTS USING SAS

- You have a data set with one dependent variable (y) and one independent variable (a).
- (a) has 4 different treatment levels (1, 2, 3, and 4).
- You want to do the following comparisons between treatment levels:
- 1)Y1-Y2
- 2) Y3-Y4
- 3) (Y1-Y2)/2 - (Y3-Y4)/2

DOING PLANNED ORTHOGONAL CONTRASTS USING SAS

- SAS INPUT:

data dataset;

input y a;

cards;

3 1

4 2

7 3

7 4

.............

proc glm;

class a;

model y = a;

contrast 'Compare 1 and 2' a 1 -1 0 0;

contrast 'Compare 3 and 4' a 0 0 1 -1;

contrast 'Compare 1 and 2 with 3 and 4' a 1 1 -1 -1;

run;

Give your dataset an informative name.

Tell SAS what you’ve inputted: column 1 is your y variable (dependent) and column 2 is your a variable (independent).

This is followed by your actual data.

“Model” tells SAS that you want to look at the effects that a has on y.

Use SAS procedure proc glm.

“Class” tells SAS that a is categorical.

Indicate “weights” (kind of like the coefficients).

Then enter your “contrast” statements.

RECOMMENDED READING

- Kirk RE. 1982. Experimental design: procedures for the behavioural sciences. Second ed. CA: Wadsworth, Inc.
- Field A, Miles J. 2010. Discovering Statistics Using SAS. London: SAGE Publications Ltd.
- Institute for Digital Research and Education at UCLA: http://www.ats.ucla.edu/stat/
- Stata, SAS, SPSS and R.

PLANNED COMPARISONS USING A t STATISTIC

- t = ∑cjYj/ √MSerror∑cj/nj
- Where c is the coefficient, Y is the corresponding mean, n is the sample size, and MSerror is the Mean Square Error.

PLANNED COMPARISONS USING A t STATISTIC

- For Example, you want to compare 2 means:
- Y1 = 48.7 and Y2=43.4
- c1 = 1 and c2 = -1
- n=10
- MSerror=28.8
- t = (1)48.7 + (-1)43.4

[√28.8(12/10) + (-12/10)]

= 2.21

CONFIDENCE INTERVALS FOR ORTHOGONAL CONTRASTS

- (Y1-Y2) – tα/2,v(SE) ≤ ψ ≤ (Y1-Y2) – tα/2,v(SE)
- For Example:
- If, Y1 = 48.7, Y2 = 43.4, and SE(tα(2),df) = 4.8
- (Y1-Y2) – SE(tα(2),df) = 0.5
- (Y1-Y2) + SE(tα(2),df) = 10.1
- Thus, you can be 95% confident that the true value of ψ is between 0.5 and 10.1.
- Because the confidence interval does not include 0, you can also reject the null hypothesis that Y1-Y2 = 0.

(Example after Kirk 1982)

TYPE I ERRORS AND ORTHOGONAL CONTRASTS

- As the number of independent tests increases, so does the probability of committing a Type 1 error.
- For Example, when α = 0.05:
- 1-(1-0.05)3=0.14 (C=3)
- 1-(1-0.05)5=0.23 (C=5)
- 1-(1-0.05)10=0.40 (C=10)

(Example after Kirk 1982)

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