**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) 1^{st} IV, b) 2^{nd} 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 2^{nd} 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

Ú*n* = #
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)

Ú*q* = #
levels of Factor *A*; *r* = # levels of Factor *B*; *n* = #
subjects in a cell.