Factorial Designs - ANOVA, MANOVA, ANCOVA, Multiple Regression

ANCOVA

- ANCOVA examines the influence of an independent variable on a dependent variable while removing the effect of the covariate factor. ANCOVA first conducts a regression of the independent variable (i.e., the covariate) on the dependent variable. -The residuals (the unexplained variance in the regression model) are then subject to an ANOVA. Thus the ANCOVA tests whether the independent variable still influences the dependent variable after the influence of the covariate(s) has been removed.

Factorial Design 2x3

- Has two independent variables, one has two levels the other IV has three levels - 6 experimental conditions

Multiple Regression

- Method for studying the relationship between a dependent variable and two or more independent variables (predictor variables) Two IVs may predict better than one Evaluating predictors jointly can answer: - Degree to which predictors are confounded or intercorrelated - After controlling for one predictor, does another IV significantly predict scores? purposes: prediction, explanation, theory building One regression coefficient for each independent variable

Purpose of Multiple Regression

- predicting values on the DV from weighted, linear combination of IVs - also explaining causal relationships - Scores on DV are predicted using multiple predictor variables - look for line of "best fit" for predicting DV - significance tests to assess how each variable contributes to the equation.

Multiple Regression Case Example

- suppose that there was a model developed for predicting graduate students grade point average. Data were collected from 50 students on the following variables: GPA, GREQ, GREV, MAT and AR. What is the DV? grade point average What are the predictor variables? GPA, GREQ, GREV, MAT and AR Do you think there is redundancy among any of the predictor variables? Yes, the two GRE's (maybe) What regression model would you choose to use?

Main Effect

- the effect each independent variable has by itself. The overall effect of the variable on the dependent variable - dependent on the amount of factors - Ex: measuring math instruction - you might have 3 levels of math instruction, but as an IV, you are just looking at the entirety of that factor. If there is a significant effect, there is the main effect of type of instruction on performance. Ex: Factorian design of 2x2 can have 2 main effects because there are two factors

Why use a factorial design

- the researcher can study the effects of two or more independent variables in combination - this is important because sometimes the effect of one variable may vary across the levels of another variable - When this happens it is called an interaction effect

Primary Reasons for MANOVA

- treatment may affect subjects in multiple ways - reduce Type 1 error - multiple on-way ANOVA's with dependent variables that are correlated or related increases the risk for Type 1 error - also decreases power - can use multiple if they are not related - Incorporates intercorrelations among DVs - Using several DVs provides more detailed and reliable information - hard to get a good assessment of a trait with just one measure - MANOVA can be a more powerful too - in some situations, more likely to detect statistical sig. differences between groups

Multicollinearity

-we desire predictors to correlate highly with DV - we do NOT desire predictors to correlate highly with each other, but they can correlate a little multicollinearity: - Highly correlated predictors will contain the same information - redundancy - R will be reduced - Interpretation becomes muddled on respective contributions of IVs - Increases variances of regression coefficients Options: remove variables or combine them

MANOVA research questions

1) What are the main effects of the independent variables? 2) What are the interactions among the independent variables? 3) What is the importance of the dependent variables? 4) What is the strength of association between dependent variables? 5) What are the effects of covariates? How may they be utilized?

Purposes of ANCOVA

1. Increase sensitivity of F ratios by removing "unwanted and predictable variance associated with the covariates." - the variable we control for is one we are not interested in. We want to look at the effects of our key variables with the effects of the unwanted variable, statistically controlled for - the effect of the covariate is partialed out of the within-groups variability - the key issue is systematic, predictable variance attributed to the covariate 2. Uses statistical matching to address non-randomly assigned group comparisons - means for each group on the DV are adjusted as if they all scored equally on the covariate 3. Include multiple DVs (MANCOVA)

Multiple Correlation or R

1. Variance predictable from the combination of predictors 2. Variance that is not predictable from the predictors - assesses the overall goodness of prediction of Y scores from X1 and X2 - null hypothesis - r = 0 - Significance test: an F ratio

Factor

Another term for independent variable Example of two variables: types of math instruction and math aptitude

Repeated Measures Design

Assigns participants to all conditions - each participant is assigned to each level of the IV (high and low medication dose) Each participant is measured after receiving each level of the IV, this is also referred to as within-subjects design (comparisons are made within the same groups of participants) Ex: In a 2x2 design, each individual would participate in all conditions. If you wanted 10 participants in each condition, a total of 10 subjects would be needed.

Sampling can affect ANOVA

Ex: Low, medium and high sex offenders have to go to individual therapy. There is a large within group difference and not a very large between group difference. Individual therapy does not work between groups, but there's a lot of variance within groups. Whenever you have a group that's too heterogeneous, like sex offenders (there are subgroups such as online offenders vs. contact offenders), which is why there is more in group variance.

Factorial Designs - Variables

In a factorial design, the researcher must manipulate as least one of the independent variables - this manipulated variables is called the active variables - experimental design The other one can be a non-manipulated variable - for example: gender, age, a personality variable, or family-related variable

Interaction Effect

Interaction effect: the effect of one IV is different at different levels of the IV The effect of one IV depends on the particular levels of the other Ex: Study looking the effect of some sleep drug (Halcion) and alcohol consumption on overall sleep time. This is a 2x2 design IVs = sleep drug and alcohol consumption DV = Overall sleep time Randomly assign participants to receive either 1 mg or 10 mg of Halcion. Randomly assign participants to different levels of the other IV; either 12 oz beer or 36 oz beer. Therefore there are 4 experimental designs - 4 groups Possible outcomes: Each IV by itself influences sleep. Participants are getting both the sleep drug and alcohol. It's possible that the two IVs are "interacting" in some way to affect sleep time. Therefore, in a 2x2 design there is only 1 interaction that can occur. For a 2x2x2 design, four interactions can occur. One interaction between factors 1 and 2. Second interaction between factors 1 and 3. Third interaction between factors 2 and 3. Then an interaction between all three factors.

ANOVA (Analysis of Variance)

Looks at the differences (variance) within groups and between groups to see if there is a significant difference Parametric statistic Extended version of a t-test Examines the means for 2 or more groups - ALWAYS 2 OR MORE Test statistics is an F ratio

Advantage and Disadvantage of MANOVA

MANOVA is useful in experimental situations where at least some of the independent variables are manipulated. It has several advantages over ANOVA. First, by measuring several dependent variables in a single experiment, there is a better chance of discovering which factor is truly important. Second, it can protect against Type I errors that might occur if multiple ANOVA's were conducted independently. Additionally, it can reveal differences not discovered by ANOVA tests. However, there are several cautions as well. It is a substantially more complicated design than ANOVA, and therefore there can be some ambiguity about which independent variable affects each dependent variable. Thus, the observer must make many potentially subjective assumptions. Moreover, one degree of freedom is lost for each dependent variable that is added. The gain of power obtained from decreased SS error may be offset by the loss in these degrees of freedom. Finally, the dependent variables should be largely uncorrelated. If the dependent variables are highly correlated, there is little advantage in including more than one in the test given the resultant loss in degrees of freedom. Under these circumstances, use of a single ANOVA test would be preferable.

ANCOVA examples

Medicine - Does a drug work? Does the average life expectancy significantly differ between the three groups that received the drug versus the established product versus the control? This question can be answered with an ANOVA. The ANCOVA allows to additionally control for covariates that might influence the outcome but have nothing to do with the drug, for example healthiness of lifestyle, risk taking activities, or age. Sociology - Are rich people happier? Do different income classes report a significantly different satisfaction with life? This question can be answered with an ANOVA. Additionally the ANCOVA controls for confounding factors that might influence satisfaction with life, for example, marital status, job satisfaction, or social support system. Management Studies - What makes a company more profitable? A one, three or five-year strategy cycle? While an ANOVA answers the question above, the ANCOVA controls additional moderating influences, for example company size, turnover, stock market indices.

Assumptions of MANOVA

Normal Distribution: - The dependent variable should be normally distributed within groups. Overall, the F test is robust to non-normality, if the non-normality is caused by skewness rather than by outliers. Tests for outliers should be run before performing a MANOVA, and outliers should be transformed or removed. Linearity - MANOVA assumes that there are linear relationships among all pairs of dependent variables, all pairs of covariates, and all dependent variable-covariate pairs in each cell. Therefore, when the relationship deviates from linearity, the power of the analysis will be compromised. Homogeneity of Variances: - Homogeneity of variances assumes that the dependent variables exhibit equal levels of variance across the range of predictor variables. Remember that the error variance is computed (SS error) by adding up the sums of squares within each group. If the variances in the two groups are different from each other, then adding the two together is not appropriate, and will not yield an estimate of the common within-group variance. Homoscedasticity can be examined graphically or by means of a number of statistical tests. Homogeneity of Variances and Covariances: - In multivariate designs, with multiple dependent measures, the homogeneity of variances assumption described earlier also applies. However, since there are multiple dependent variables, it is also required that their intercorrelations (covariances) are homogeneous across the cells of the design. There are various specific tests of this assumption.

Simple Regression

One dependent variable predicted by on independent variable one regression coefficient

One-Way ANOVA

One-way ANOVA example: As a crop researcher, you want to test the effect of three different fertilizer mixtures on crop yield. You can use a one-way ANOVA to find out if there is a difference in crop yields between the three groups. Use a one-way ANOVA when you have collected data about one categorical independent variableand one quantitative dependent variable. The independent variable should have at least three levels (i.e. at least three different groups or categories). ANOVA tells you if the dependent variable changes according to the level of the independent variable. For example: Your independent variable is social media use, and you assign groups to low, medium, and high levels of social media use to find out if there is a difference in hours of sleep per night. Your independent variable is brand of soda, and you collect data on Coke, Pepsi, Sprite, and Fanta to find out if there is a difference in the price per 100ml. You independent variable is type of fertilizer, and you treat crop fields with mixtures 1, 2and 3 to find out if there is a difference in crop yield. The null hypothesis (H0) of ANOVA is that there is no difference among group means. The alternate hypothesis (Ha) is that at least one group differs significantly from the overall mean of the dependent variable. If you only want to compare two groups, use a t-test instead.

Independent Groups Design

Participants are randomly assigned to various conditions (each participant is only in one group) - this is referred to was between-subjects design, comparisons made between groups of participants Ex: In a 2x2 design, there are 4 conditions Different groups of participants will be assigned to each of the four conditions

Multiple Correlation Coefficient (R) and Coefficient of Multiple Determination (R^2)

R = the magnitude of the relationship between the dependent variable and the best linear combination of the predictor variables R^2 = the proportion of variation in Y accounted for by the set of independent variables (X's)

Why not run multiple t-tests?

Risk of inflated Type 1 error - the more t-tests the more room for error - 3-4 tests is probably fine, but more than that is not

MANOVA example

Say you wanted to know if going to high-school or college affected how men and women did in their careers. Your independent variables are gender of individual and the amount of education achieved. You measure how men and women did in life in multiple ways: income, number of promotions gained, and a test of overall job happiness of each individual (these are your dependent variables). This is an example of a study that could be analyzed using a MANOVA.

Factorial Designs 2x2

Simplest factorial design has two independent variables, each having two levels 4 experimental conditions Ex: Factor 1: Gender, with 2 levels - female - male Factor 2: Depression medication dosages with 2 levels - 20mg - 40mg

Different Regression Models

Simultaneous: all independent variables entered together Stepwise: independent variables entered according to some order - Theory of what accounts for the most - some order of variables that accounts for the most variability of the DV - by size or correlation with DV - in order of significance Hierarchical: independent variables entered in stages - theory - entering IV's in stages - I think IQ accounts for most so I'm going to enter that first, and so on - hierarchy base don the order of which you think the IV's account for the most variability

Uses of ANCOVA

The ANCOVA is most useful in that it (1) explains an ANOVA's within-group variance, and (2) controls confounding factors. Firstly, as explained in the chapter on the ANOVA, the analysis of variance splits the total variance of the dependent variable into: 1. Variance explained by the independent variable (also called between groups variance) 2. Unexplained variance (also called within group variance)

Assumptions of ANOVA

The assumptions of the ANOVA test are the same as the general assumptions for any parametric test: Independence of observations: the data were collected using statistically-valid methods, and there are no hidden relationships among observations. If your data fail to meet this assumption because you have a confounding variable that you need to control for statistically, use an ANOVA with blocking variables. Normally-distributed response variable: The values of the dependent variable follow a normal distribution. Homogeneity of variance: The variation within each group being compared is similar for every group. If the variances are different among the groups, then ANOVA probably isn't the right fit for the data.

Experimental design - assigning participants

There are two ways of assigning participants which determine the type of experimental design and kind of comparisons made 1. independent groups design - between subjects design 2. Repeated Measures Design - within subjects design

Multivariate Analysis of Variance (MANOVA)

Trying to see variables and their effects on several-outcomes simultaneously Can have two or more DVs and can have more than one IV Test - multivariate F-ratio

Example of ANOVA

Want to see the difference between three groups of offenders, were going to give an intervention of group therapy. We are going to be specific and say these are groups of sex offenders, low risk, medium risk and high risk. Therefore there is one factor with three levels. Going to see if there's a difference between those groups and recidivism. We will see if there's a difference between by having them attend group therapy for 12 months. After running an ANOVA, we see a large between group difference that demonstrated that the high risk group was most affected.

When to use a two-way ANOVA

You can use a two-way ANOVA when you have collected data on a quantitative dependent variable at multiple levels of two categorical independent variables. Fertilizer types 1, 2, and 3 are levels within the categorical variable fertilizer type. Planting densities 1 and 2 are levels within the categorical variable planting density. You should have enough observations in your data set to be able to find the mean of the quantitative dependent variable at each combination of levels of the independent variables. Both of your independent variables should be categorical. If one of your independent variables is categorical and one is quantitative, use an ANCOVA instead. Ex: Looking at how gender (one IV with two levels: male/female) and type of intervention (small process group, one-to-one, small psychoed group) interact on sense of community score (DV) - has to be two IVs with multiple levels and one DV

F Statistic

a ratio of two measures of variance: (1) between-groups variance, which indicates differences among sample means, and (2) within-groups variance, which is essentially an average of the sample variances If there is a larger difference between the groups then the F value is larger, and we say there is a significant difference ANOVA uses the F-test for statistical significance. This allows for comparison of multiple means at once, because the error is calculated for the whole set of comparisons rather than for each individual two-way comparison (which would happen with a t-test). The F-test compares the variance in each group mean from the overall group variance. If the variance within groups is smaller than the variance between groups, the F-test will find a higher F-value, and therefore a higher likelihood that the difference observed is real and not due to chance.

Design of Multiple Regression

one dependent variable (criterion two or more independent variables (predictor variables) Sample size: >/= 50 (at least 10 times as many cases as independent variables)

When to use an ANCOVA

- to correct for pre-existing groups that are not equivalent on some characteristics - to analyze pretest/post-test scores - to reduce error variance due to individual difference - Another variable (covariate) is confounded with one of your predictors - the covariate is highly predictive of the outcome variable. Including it in your model, then, accounts for unexplained variance

What could cause variance between groups

- treatment effect - chance - individual differences - experimental error

One-way vs. Two-way ANOVA

A one-way ANOVA uses one independent variable, while a two-way ANOVA uses two independent variables.

Two-Way ANOVA

A two-way ANOVA is used to estimate how the mean of a quantitative variable changes according to the levels of two categorical variables. Use a two-way ANOVA when you want to know how two independent variables, in combination, affect a dependent variable. Example You are researching which type of fertilizer and planting density produces the greatest crop yield in a field experiment. You assign different plots in a field to a combination of fertilizer type (1, 2, or 3) and planting density (1=low density, 2=high density), and measure the final crop yield in bushels per acre at harvest time. You can use a two-way ANOVA to find out if fertilizer type and planting density have an effect on average crop yield.

Factorial Designs

Factorial designs involve two or more independent variables All levels of each independent variable are combined with all levels of the other independent variable

Multiple Regression Assumptions

Independence: the scores of any particular subject are independent of the scores of all other subjects Normality: int he population, the scores on the dependent variable are normally distributed for each of the possible combinations of the level of the X variables, each of the variables if normally distributed Homoscedasticity: in the population, the variances of the dependent variable for each of the possible combinations of the levels of the X variables are equal Linearity: In the population, the relation between the dependent variable and the independent variable is linear when all the other independent variables are held constant.

Level of a factor

One of the form or variants of an independent variable. Each IV has 2 or more levels Example: The two levels of math instruction (factor 1) are direct and indirect The two levels of math aptitude (factor 2) are high and low