Good strategic decision making requires first and foremost, high quality information. For managers and other decision makers it is therefore crucial to understand the quantitative and statistical methods, and their drawbacks, that are so often used to generate the information they are provided with...
What is a “model”?
- A simplified description of reality
- A visual representation of relations between theoretical constructs and variables
Moderating variable
- One variable moderates the relationship between two other variables
- It can either strengthen or weaken the existing relationship
- Communication skills moderate relationship between boring teacher and bored kids
- Provides nuance to research
- It is about change and about the relationship (interaction) between PVs
- The effect of one PV on the OV is moderated (depends on) another PV
- Can be positively or negatively moderating
- The moderator is only relevant if it happens together with the PV
- How sweet your coffee (OV) is depends on how much sugar you put in (PV)
moderated (MV) by how much you stir it
Mediating variable
- One variable mediates the relationship between the two other variables
- “Vehicle” variable: explains the relationship between the two other variables
- Slide quality mediates relationship between boring teacher and bored kids
- Some can be both mediating and moderating
How do you deal with multiple groups of data?
- We want to explain variability in the OV
- “Between groups” variability is identified as the variability explained by the model
- “Within groups” variability is variability not explained by the model
- ANOVA shows how group variability explains variability in the OV
- Comparing the means between groups shows “between group” variability
- ANOVA tests for the differences in the mean between groups
- ANOVA needs a quantitative OV and a categorical PV
- ANOVA decomposes total variance into variance explained by the model and
residual variance
ANOVA: (ANalysis Of VAriance)
- Compares the variability between groups with the variability within the groups
- How much of the outcome variable is explained by the predictor variable?
- Measurements of variability (how values differ in data) between groups
- Comparing differences between several means
, - Can go “one way” (one PV) or “two way” (more than one PV)
- ANOVA translates total variance into:
o Variance explained by the model
o Variance which is residual
How does ANOVA work?
- A one-way ANOVA can compare 2 (independent, “between-subject”) groups
- How much of the variability in our outcome is explained by our PV?
- Compares the variability between the groups against the variability within the groups
- Some variability can be explained by the PV
- But there is always residual variability that is unexplained
- ANOVA breaks down different measures of variability with sums of squares
- Helps us test if the mean scores of the groups are statistically different
ANOVA assumptions:
- When the OV is Quantitative
- When the PV is Categorical
- When there are more than 2 groups of PVs
- When variance is homogenous across groups
- When residuals are normally distributed
- When groups are roughly equally sized
- When data is “between subjects” (subjects can only be in one group at a time)
How to check for “homogeneity of variances”? = Levene’s test!
1) Find the table “Test of homogeneity of Variances”
2) Check the “sig”
a. Higher than 0.05? = group means homogenous, ANOVA can proceed
b. Significance? = group means heterogeneous, ANOVA can’t proceed
ANOVA Hypothesis:
H0 = no difference in μ’s between groups (μ1 = μ2 = μ3…)
H1 = at least two group μ’s are different (μ1 ≠ μ2 ≠ μ3…)
Conclusion: If the p < .05, then there is a statistically significant difference between at least
two of the group means and we reject the H0.
,ANOVA 1: calculate SST, the Total Sum of Squares
The step to take before you can run an ANOVA test:
- To find the total amount of variation within data, we calculate the difference between
each observed data point and the grand mean (Grand Mean = ybar = ȳ)
- Simplest model to fit a set of data: the mean of the PV and OV
- Square the differences, add them up = SST (Sum of Squares) (or Grand Variance)
- s2 (variance) = SS (Sum of Squares) ÷ (N – 1)
- SS = s2 x (N – 1)
Calculate the SST:
1) Start with the observed scores
2) Find the mean of all observed scores
3) Subtract observed means from grand mean (y - ȳ)
4) Square the result (y - ȳ)2
5) Add up everything Σ (y - ȳ)2
6) Multiply each result by the number of participants (“n”) Σn (y - ȳ)2
Sum of Squares formula:
Sum of Squares does not explain variance within each group; for this we need “residual”
ANOVA 2: Calculate the Sum of Squares “Between” and the Sum of Squares “Within”
- How much of the total variance is found between the total means of group 1
compared with group 2, compared with group 3 etc? = MODEL
- How much is found within group 1 or within group 2 etc? = RESIDUAL
What is “analysis of variance”?
- For a key outcome variable which is quantitative, we want to understand the spread
- High variance = broad range of responses, inconsistent values
- Low variance = narrow range of responses, scores that are similar to each other
Sum of Squares
- The Total Sum of Squares is the total variance across all groups (between-groups)
- Total Sum of Squares (SST) is made up of the Model Sum of Squares (SSM) and
Residual Sum of Squares (SSR)
o SSM explains variation between the various groups, explained by the model
o A low SSM means the model is a good fit
o An SSM of “0” means that there is no difference in the means of all groups
(but there can be high variability within the groups themselves)
o SSR explains other, random, unexplained variation
SStotal = SSmodel + SSresidual
Residual Sum of Squares = RSS or SSR
- How much of the variation cannot be explained by the model?
- What is the amount of “random” variation in the sample (individual height, weight)?
- What is the difference between what the model predicts and what was actually
observed?
SSR = Σ (SSGROUP 1 + SSGROUP 2 + SSGROUP 3…)2
, SSR = Σ (VAREACHGROUP) x (n-1)
We come out with a percentage score:
If the SSR = 0%, then all groups have the same mean: our model, that PV influences
OV, means nothing
This means 95% of the variability within scores is due to the PV
“factor” is a PV
“levels” are PV categories (groups)
ANOVA in SPSS
F-Ratio
- F-ratio measures the improvement due to fitting the model
- Compares the group means versus the grand mean of scores for all participants
- Compares this against the error remaining in the model, which is the difference
between the actual scores and the respective means of the groups
- Ratio of explained variance relative to unexplained variance
- How good a test model is compared to the error within that model
- Divide the model mean squares (MSM) by the residual means square (MRR)
- If the value is less than 1, the effect is non-significant
- A good model has a large F-Ratio (greater than 1) because the “model” should be
larger than the “error”
F = MSM or systematic variation
MSR unsystematic variation
FRATIO = explained variability = between group variability
unexplained variability within group variability
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