Question

1) In MANCOVA, Independent variables, Dependent variables and covariate must each confirm to a specific level of measurement. List the correct level of measurement for each of the variable listed Independent variables, Dependent variables and covariate

2) Differentiate between a Total Effect, Direct Effect and Indirect Effect within the decomposition of effects approach for determining statistical mediation

3) Describe the similarities and differences between simple mediation and moderation atleast four points

4) When and why would one choose to interpret pillai's Trace multivariate test statistics over wilks Lambda?

5) What is bootstrapping and why is it used in statistical analyses involving mediation?

6) How does MANOVA differe from ANOVA. When would you select to run a MANOVA over an ANOVA and why would MANOVA be advantageous in such situations.

7) Define a coavriate. How do you choose a covariate?

Answer 1

a)

Multivariate analysis of covariance (MANCOVA) is a statistical technique that is the extension of analysis of covariance (ANCOVA). Basically, it is the multivariate analysis of variance (MANOVA) with a covariate(s).). In MANCOVA, we assess for statistical differences on multiple continuous dependent variables by an independent grouping variable, while controlling for a third variable called the covariate; multiple covariates can be used, depending on the sample size.

Level and Measurement of the Variables:

In MANCOVA assumes that the independent variables are categorical and the dependent variables are continuous or scale variables. Covariates can be either continuous, ordinal, or dichotomous.

c) Similarities and differences between simple mediation and moderation:

Both mediation and moderation have to do with checking on how a third variable fits into that relationship. For the purposes of understanding these two concepts, this is where the similarities end.

Moderation is a way to check whether that third variable influences the strength or direction of the relationship between an independent and dependent variable. An easy way to remember this is that the moderator variable might change the strength of a relationship from strong to moderate, to nothing at all. It is almost like a turn dial on the relationship; as you change values of the moderator, a statistical relationship that you observed before might dissolve away. For example, if you expected that the length of time studying related to the grades on a calculus test, you would probably be right. Let’s say there is a strong relationship between time spent studying and grades. However, that relationship may not hold true across the board; something like grade level might be a possible moderator. If you switch the value of this moderator from college student to elementary school student, that relationship is not likely to hold up. No amount of studying is likely to help a second grader an A on a calculus exam, but for a college student, study time will matter a great deal. Mediation is a little more straightforward in its naming convention. A mediator mediates the relationship between the independent and dependent variables – explaining the reason for such a relationship to exist. Another way to think about a mediator variable is that it carries an effect. In a perfect mediation, an independent variable leads to some kind of change to the mediator variable, which then leads to a change in the dependent variable. However, in practice, the relationships between the independent variable, mediator, and dependent variable are not tested for causality, just a correlational relationship.

The purpose of mediation analysis is to see if the influence of the mediator is stronger than the direct influence of the independent variable. An obvious real-life mediator is temperature on a stove. Water will not start to boil until you have turned on your stove, but it is not the stove knob that causes the water to boil, it is the heat that results from turning that knob. To test something like this, we could check to see how tightly correlated the knob being turned is to the water’s state . For the first few minutes there would be no effect, so we can treat that as a weak correlation. Compared to the relationship between the temperature or stove top and the state of the water, we can see that it is actually the temperature of the stove (the mediator) that is causing the water to boil, not just the action of turning a knob (the independent variable). Comparing the strength of these effects gives you insight into what is really carrying the effect on the water (the dependent variable).

d)  Pillai’s Trace:

Pillai’s trace is used as a test statistic in MANOVA and MANCOVA. This is a positive valued statistic ranging from 0 to 1. Increasing values means that effects are contributing more to the model; you should reject the null hypothesis for large values. Pillai’s is one of several test statistics used in MANOVA; Other commonly used tests include: Hotelling’s T², Roy’s largest rootRoy’s Largest Root (Criterion): Definition and Wilks’ Lambda).

When to use this test:

This test is considered to be the most powerful and robust statistic for general use, especially for departures from assumptions. For example, if the MANOVA assumption of homogeneity of variance-covariance is violated, Pillai’s is your best option. It is also a good choice when you have uneven cell sizes or small sample sizes (i.e. when n is small). However, when the hypothesis degrees of freedom is greater than one, Pillai’s tends to be less powerful than the other three. If you have a large deviation from the null hypothesis or the eigenvalues have large differences, Roy’s Maximum Root is a far better option (Seber 1984).

Formula:

The formula for Pillai’s trace is:


The general steps for calculating the test statistic are:

1. Divide each eigenvalue by 1 + the characteristic root.
2. Sum these ratios.

It’s extremely rare that you would want to actually perform these calculations by hand; Most statistical packages will perform the calculations for you as part of MANOVA or MANCOVA. The output will return a Pillai’s trace value, associated F-statistic and a p-value. In general, small p-values (below .05) mean that Pillai’s returned a significant result (that there is a difference between the levels of independent variable you are looking at).

e)  Bootstrapping

Bootstrapping is any test or metric that relies on random sampling with replacement. Bootstrapping allows assigning measures of accuracy (defined in terms of bias, variance, confidence intervals, prediction error or some other such measure) to sample estimates. This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods. Generally, it falls in the broader class of resampling methods. Bootstrapping is the practice of estimating properties of an estimator (such as its variance) by measuring those properties when sampling from an approximating distribution. One standard choice for an approximating distribution is the empirical distribution function of the observed data. In the case where a set of observations can be assumed to be from an independent and identically distributed population, this can be implemented by constructing a number of resamples with replacement, of the observed dataset (and of equal size to the observed dataset).

Bootstrapping is typically used because the indirect effect is itself often nonnormal, which invalidated analytically-derived SEs and thus p-values and their corresponding 95% CIs. Bootstrap CIs have become the standard solution. However, Monte Carlo CIs can work reasonably well and the 95% intervals from a Bayesian analysis done with MCMC are also fine for nonnormal effects.

f)

The ANOVA method includes only one dependent variable while the MANOVA method includes multiple, dependent variables.

ANOVA uses three different models for experimentations; random-effect, fixed-effect, and multiple-effect methods to determine the differences in means which is its main objective while MANOVA determines if the dependent variables get significantly affected by changes in the independent variables. It also determines the interactions taking place amongst dependent variables and determines the interactions taking place amongst independent variables too.

Multivariate analysis of variance (MANOVA) is simply an ANOVA with several dependent variables. That is to say, ANOVA tests for the difference in means between two or more groups, while MANOVA tests for the difference in two or more vectors of means.

A multivariate analysis of variance (MANOVA) could be used to test this hypothesis. Instead of a univariate F value, we would obtain a multivariate F value (Wilks' λ) based on a comparison of the error variance/covariance matrix and the effect variance/ covariance matrix. Although we only mention Wilks' λ here, there are other statistics that may be used, including Hotelling's trace and Pillai's criterion. The "covariance" here is included because the two measures are probably correlated and we must take this correlation into account when performing the significance test. Testing the multiple dependent variables is accomplished by creating new dependent variables that maximize group differences. These artificial dependent variables are linear combinations of the measured dependent variables.

g)  Covariate

In general terms, covariates are characteristics (excluding the actual treatment) of the participants in an experiment. If you collect data on characteristics before you run an experiment, you could use that data to see how your treatment affects different groups or populations. Or, you could use that data to control for the influence of any covariate.

Covariates may affect the outcome in a study. For example, you are running an experiment to see how corn plants tolerate drought. Level of drought is the actual “treatment”, but it isn’t the only factor that affects how plants perform: size is a known factor that affects tolerance levels, so you would run plant size as a covariate.

A covariate can be an independent variable (i.e. of direct interest) or it can be an unwanted, confounding variable. Adding a covariate to a model can increase the accuracy of your resul