Experiments with Multiple Independent Variables
Experimental Psychology
Lecture, Chapter 12
Factorial Designs
ÚRarely in nature is one’s behavior affected by a single influence. More likely will behavior be affected by multiple influences.
ÚFactors, or IV’s, can be studied in groups using a factorial design.
ÚAs opposed to increasing the number of levels, or groups, within one IV, factorial designs involve multiple IV’s.
Naming/Identifying Designs
ÚOne-way ANOVA refers to 1 IV, two-way ANOVA refers to 2 IVs, etc…
ÚIf you use numerical identification with your design, more information is given.
–The # of digits reflects the # of IVs.
–The digits themselves reflect # levels within each ANOVA.
–For instance, a 2 x 2 ANOVA indicates 2 IV’s, both having 2 levels.
Factorial designs show:
ÚTrends in the IV’s independently, called main effects (the sole effect of one IV).
–Effects of all IVs independent of one another
ÚJoint effects of the IV’s together, called interaction effects (the simultaneous effect of more than one IV).
–Effects of one IV dependent on the level of the other IV.
Group Assignment
ÚIndependent – random assignment (sometimes referred to as “between-groups” designs)
ÚCorrelated groups (sometimes referred to as “within-groups” designs)
–Matched pairs or sets
–Repeated measures
–Natural pairs or sets
ÚMixed assignment – combination of random and nonrandom assignment with at least 1 variable in both conditions
–Usually 2 independent groups measured repeatedly
Why use multiple group designs?
Ú2 x 2 factorial designs are ideal for preliminary investigations of two IVs.
ÚMultiple group designs (ex. 3 x 2) provide information beyond what would be learned from conducting separate experiments (ex. conducting three 2-group experiments)… interaction effects.
ÚMultiple group designs are able to answer more complex research questions and a greater number of them in one experiment.
Different Factorial Designs
ÚVarying amounts of an IV – when adding a level to a multiple-groups design, you’re only adding 1 group; to a factorial design, you’re adding as many groups as you have IVs.
ÚFactorial designs can be used with non-manipulated variables (resulting in ex post facto research).
ÚMore than 2 IVs can be used, increasing the complexity of the design, the number of possible interactions, and the difficulty in interpretation.
–Rather than just one interaction between the 2 IVs, you must evaluate: AB, AC, BC, and ABC.
Statistical Analysis
ÚUsing factorial ANOVAs:
–Variability is still partitioned into 2 sources, treatment and error.
–More sources of variability (ex. In a 2 IV experiment, variability will result from both treatment and error from 3 sources: a) 1st IV, b) 2nd IV, and c) interaction)
–Treatment variability in DV is that which is due to IVs (between groups variability)
–Error variability in DV is that which is due to factors other than the IVs, such as individual differences and measurement error (within groups variability)
Sources of Variability
Whereas with a one-way ANOVA, we had the following sources of variability,
F = between groups variability
within groups variability
With a two-way ANOVA, we add sources of variability:
Fa = IV a var. Fb = IV b var. Faxb = interaction var.
error var. error var. error var.
Understanding Interactions
ÚSynergistic effects are dramatic consequences that occur when you combine 2+ substances, conditions, or organisms. The effects are greater or less than what is individually possible.
ÚA significant interaction means that the effects of the various IV’s are not straightforward and simple; it means that one IV depends on the specific level of the other IV.
ÚFor this reason, we virtually ignore our IV main effects when we find a significant interaction.
ÚGraphing helps make sense of interactions (DV on y axis, IV on x axis, and 2nd IV depicted with lines on the graph).
2-Way ANOVA for Independent Samples
In computer analysis, you will receive the following information:
ÚDescriptive statistics for all groups
ÚMain effects of the IVs, in terms of F statistic and significance level.
ÚInteraction effects of IVs, which if significant render the main effects moot.
ÚWithin groups effects, or error term
ÚNote: if probabilities of the effects due to chance are above 5% (p = .05) but below 10% (p = .1), results should be mentioned in terms of marginal significance.
ÚRemember that as probabilities of chance increase, so does your risk of making a Type I error
2-Way ANOVA for Correlated Samples
In correlated samples, correlated groups (usually repeated measures) need to be used in both IVs. The computer analysis will yield the following information (see p. 320):
ÚDescriptive statistics for all groups
ÚMain effects of the IVs, in terms of F statistic and significance level (usually higher).
ÚInteraction effects of IVs, which if significant render the main effects moot (usually higher).
ÚWithin groups effects, or error term, sometimes indicated as residuals (usually lower).
ÚResults are written in the same way; remember that when interaction effects are significant, a graph should be plotted.
2-Way ANOVA for Mixed Samples
In mixed samples, correlated groups need to be used for one IV and independent groups for the other IV. The computer analysis will yield the following information (see p. 323):
ÚDescriptive statistics for all groups
ÚBetween subjects effects, which will include the effect of the independent groups IV, as well as error.
ÚWithin subjects effects, which will include the effect of the correlated groups IV and the interaction effect, as well as error.
ÚResults are written in the same way; remember that when interaction effects are significant, a graph should be plotted.
ÚAlso, post hoc tests are used with 3+ levels to an IV to determine where the significant difference between group means is located.
Reviewing Research Steps
ÚAfter preliminary study, decide on the design for the next, more involved, study with 2+ IV’s.
ÚDefine DV.
ÚDecide on assignment method.
–Larger samples can use random assignment to all groups; in such cases, use 2-way ANOVA for independent samples.
–Smaller samples can use the same participants in all groups (or correlate participants); use 2-way ANOVA for correlated samples.
–In some cases, mixed methods can be used (random assignment to levels of one IV and use the same participants, or correlate, in all levels of the second IV participants); use 2-way ANOVA for mixed samples.
ÚMake conclusions based on statistical results.
df for 2-way Independent Samples ANOVA
ÚTreatment Effect (t): Df(t) = q – 1
ÚGender Effect (g): Df(g) = r – 1
ÚTreatment x gender interaction (tg):
–Df(tg) = (q – 1)(r – 1)
ÚWithin group (error) (w): Df(w) = qr (n – 1)
ÚTotal = Df (T) = qrn – 1
Ún = # participants in a cell; q = # levels of factor t; r = # levels of factor g.
df for 2-way Correlated Samples ANOVA
ÚTreatment: # treatment groups – 1 = t
ÚGender: # gender groups – 1 = g
ÚInteraction: t x g
ÚResidual: (total # groups – 1)(# in each group – 1)
df for 2-way Mixed Samples ANOVA
ÚBetween Subjects
A q – 1
–Error q(n – 1)
ÚWithin Subjects
B r - 1
–AB (q – 1)(r – 1)
–Error q(n – 1)(r – 1)
Úq = # levels of Factor A; r = # levels of Factor B; n = # subjects in a cell.