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Two-way ANOVA

By Lawrence Morgan,2014-06-18 15:46
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Two-way ANOVA

9 Introduction to Factorial Experiments

     Two-way ANOVA

    Example - Capsule Dissolving Experiment (Capsule.JMP)

    In this experiment researchers are interested in studying the effect of two factors or treatments on the time to begin dissolving a capsule which is recorded as the time until bubbles first appear (seconds). The factors of interest to the researchers are digestive juice type - gastric or duodenal (Factor A) and capsule type - C or V (Factor B).

    To conduct the experiment 5 capsules of each type are randomly assigned to each juice giving us 5 observations or replicates for each of the four treatment combinations

    (Gastric & C, Gastric & V, Duodenal & C, Duodenal & V). The data obtained from the experiment are shown below:

     Capsule Type

    Juice Type Means C V Type of Digestive Juice

     39.5 47.4

     45.7 43.5 Y45.7149.8 39.8 Gastric 50.2 36.1

    63.8 41.2

     Y49.8Y41.61112

     33.5 44

     36.7 41.2 Y40.2242 47.3 Duodenal 38.1 45.3

    31.2 42.7

     Y36.3Y44.12122

    Capsule Type Means Grand Mean Y43.05Y42.8512 X42.95

    We can construct plots to visualize the effects of each factor.

     Digestive Juice Capsule Type

     By plotting the mean time until By plotting the mean time until bubbles for both digestive juices, bubbles for both capsule types we see we can see that mean dissolution that the mean dissolution times for

    time for duodenal juice is slightly the capsule types are approximately

    smaller than that for gastric (about equal.

    5 seconds). 127

    Our preliminary conclusions would be first that fluid type has a small effect on the dissolution time with duodenal juice dissolving capsules about 5 seconds quicker on average, and secondly that capsule type has little or no effect.

These conclusions are completely WRONG!! Why?

    When considering the effect of two factors on the response we cannot do so marginally, i.e. individually. It is possible, for example, that the effect of digestive juice is not the same for both capsule types. If we consider the means for each of the treatment

    combinations above we see that for type C capsules the duodenal juice dissolves the capsule quicker, while the exact opposite is true for type V capsules, gastric juice dissolves the capsules faster.

    A better display shows the means for each treatment combination. Here we have a separate profile for each digestive juice showing how the capsule effect depends on the type of digestive juice we are using. This is what we call an interaction.

Questions of Interest in Two-way ANOVA:

     1) Is there a significant interaction between the two factors being studied?

     This question needs to answered first, because if we conclude there is a

     significant interaction then both effects are important and there effects can

     not be discussed individually. If we conclude there isn’t a significant

     interaction between the factors being studied then we can test the effects

     individually.

     2) Is there a significant Factor A effect?

     3) Is there a significant Factor B effect?

    As always it is important to quantify any significant differences using pair-wise comparisons and CI’s for the differences in the population/treatment means.

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Analysis in JMP

    To fit the two-way model for these data select Fit Model from the Analyze menu and put

    the response Time to Bubbles in the Y box and then highlight both Fluid & Capsule and select Full Factorial from the Macros pull-down menu as shown below.

Then click Run Model to obtain the results on the next page.

    These sections of output can be shut

    off as our interest is in primarily

    identifying which effects are

    significant. These results are in the

    Effect Tests box.

    The Fluid*Capsule interaction is

    significant (p=.0049), so we know

    both fluid and capsule type

    significantly effect the response.

    The p-values for the effects suggests that the Fluid*Capsule interaction is significant (p = .0049), which implies the main effect tests for Fluid and Capsule are of little interest.

    It is interesting to note that the main effect of Capsule is not significant (p = .9361). This

    happens because the presence of the Fluid*Capsule interaction "masks" the main effect of

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    capsule as we have seen in marginal effect plots above. The main effect of fluid is only partially masked by the Fluid*Capsule interaction and so it still tests as significant.

    Because the interaction is significant we can ONLY MEANINGFULLY COMPARE LEVELS OF ONE FACTOR WHILE HOLDING THE OTHER FACTOR FIXED! For

    example, in this experiment we can compare juice types for a given capsule type or conversely we can compare capsule types for a given digestive juice type. To do this select LSMeans Tukey HSD from the Fluid*Capsule interaction pull-down menu.

Results of the treatment mean comparisons are shown below.

    Here we see that Gastric,C and Duodenal,C mean dissolution times significantly differ. In particular we estimate that the type C capsules in gastric fluid take between 3.57 and

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    23.429 seconds longer to dissolve on average. In contrast type V capsules appear to dissolve equally well in either digestive juice.

If the interaction between the two factors is NOT significant we can use the Tukey’s

    procedure to compare the means across the levels of each factor individually. This means that we can use Tukey’s pair-wise comparisons to compare the mean response across the

    levels of factor A and factor B individually.

Checking Two-way ANOVA Assumptions (Normality and constant variance)

Assumptions:

     1. The observations between and within the treatment combinations are

     independent.

     2. The response is normally distributed for each treatment combination.

     3. The variance of the response is the same for each treatment combination.

To check the constant variance assumption we can examine the residuals plotted vs. the

    fitted values and each factor. The fitted values are simply the observed mean response at

    each of the four treatment combinations and the residuals are the deviations from the treatment combination means. The spread of the residuals, i.e. the spread of the observed response values about their respective treatment combination means, should be uniform indicating constant response variation for the different treatment combinations.

    A plot of the residuals vs. the fitted values is given each time we fit a model in JMP. The resulting plot is shown below:

    There appears to be a potential outlier in this plot, otherwise this plot looks fine.

To examine the normality assumption we assess the normality of the residuals. Save the

    residuals to the spreadsheet as shown below and use Analyze > Distribution to examine

    them. With the exception of two mild outliers, normality seems satisfied.

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    STATISTICAL DETAILS (FYI, equal sample size case only)

Two-way ANOVA Model

     Y(;;;(,!); i1,...,a j1,...,b k1,...,nijkijijijk

where,

    th k observed response value when level i of factor A and level j of factor B is used. Yijk

     effect due to the fact level i of Factor A was used. i

     effect due to the fact level j of Factor B was used. j

    thth effect due to the interaction of i level of Factor A and the j level of Factor B. (,!)ij

    ththe random error, represents the variation in the response values when the i level of Factor A and ijkththe j level of Factor B are used.

    2We assume that , i.e. the errors are normal and their variation is constant. ~N(0,)ijk

    See your text for formulae used to estimate these quantities and those used to test the

    hypotheses. The three questions of interest in a two-way ANOVA can be formulated in

    terms of these parameter values.

    1. For testing the interaction between Factors A and B we have:

    ,!H:()0 for all treatment combinationsoij H:(,!)0 for all treatment combinationsaij

    2. For testing the Factor A effect we have:

    H:0 for all ioi H:0 for all iai

    3. For testing the Factor B effect we have:

    H:0 for all joj H:0 for all jaj

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    As in one-way ANOVA the test procedures decomposes total response variation into components that measure how much variation in the response is due to Factor A, Factor B, the interaction between Factors A & B, and random error.

    SSSS;SS;SS;SS Sum of Squares: TotalABABError

    N1(a1);(b1);(a1)(b1);ab(n1)Degrees of Freedom:

SUM OF SQUARES FORMULAE:

    aabnb222 = + + nb(XX)(XX)an(XX)?????iijkj1???1111iijkj

    ababn22 + n(XXX;X)(XX)?????ijijijkij???111i??j11ijk

MEAN SQUARES (measures of variation)

    The mean square for an effect is the effect sum of squares divided by the degrees of freedom.

    SSeffectMS effectdfeffect

    2When the null hypothesis of “no effect” is true the mean squares are all estimates of ,

    the common response variance for all treatment combinations. If there is a significant effect then we expect the ; (within treatment combination variation). MS??MSeffectError

Testing Effect Significance

    For testing the main effects (A & B) and the interaction effect (AB) we simply compare

    the size of the to the . If the >> we have evidence that MSMSMSMSeffecteffectErrorError

    the effect is significant. If then we have little evidence that the effect is MS~MSeffectError

    significant. This is analogous to the comparison of the between group variation to the within group variation in One-way ANOVA.

    To compare the mean squares we use the ratio, which has an F-distribution.

    MSeffectF F-distribution (numerator df = df for the effect , denominator df = df for error) ~oMSError

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     >> 1 will lead to the conclusion that the effect in question significantly impacts the Fo

    response. Large values lead to small p-values which support effect significance. Fo

Example 2 Comparing the Effectiveness of Three Forms of

     Psychotherapy for Alleviating Depression

    Suppose that a clinical psychologist is interested in comparing the relative effectivenss of three forms of psychotherapy for alleviating depression. Fifteen individuals are randomly assigned to each of three treatment groups: cognitive-behavioral, Rogerian, and assertiveness training. The Depression Scale of MMPI serves as the response. The psychologist also wished to incorporate information about the patients severity of depression, so all subjects in the study were classified as having mild, moderate, or severe depression. Thus we have two factor of interest in this study: the treatment they received and the initial severity of their depression. It is possible some forms of therapy may be more effective for certain levels of depression so a two-way ANOVA would be an appropriate method of analysis. The results are presented in the table below.

Therapy Mild Moderate Severe

    Cognitive-Behavioral 41 51 45

    (CB) 43 43 55

    50 53 56

    54 60

    46 58

    62

    62

Rogerian (R) 56 58 59

    47 54 55

    45 49 68

    46 61 63

    49 52

    62

Assertiveness Training 43 59 55

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(A) 56 46 69

    48 58 63

    46 54 56

    47 62

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    The data in JMP is entered as shown on the

    left. The first column contains the Therapy

    grown, the second column contains the

    Degree of Severity of their depression, and

    the last column denotes the MMPI

    Depression score.

    Data File: MMPI Depression

    In JMP select Analyze > Fit Model and set up the dialog box as shown below. First highlight both Therapy and Degree of Severity by holding down the key and

    select Full Factorial from the Macro pull-down menu as shown below.

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The output is shown on the next page.

    Test results:

    Interaction

    Therapy

    Degree of Severity

Interaction Plot

    The interaction plot shown on the

    left shows no signs of non-

    parallelism and hence interaction,

    and the p-value in the ANOVA

    table suggests we have absolutely

    no evidence for its significance

    (p=.9845).

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