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Summary Applied Multivariate Data Analysis - Week 3
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Summary of Field's chapter 12, 13 and 14
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1 vérifier
Par: rabuca • 2 année de cela
mistakes on page 9
Par: galinajimberry • 2 année de cela
hi, I'm sorry to hear that, could you tell me which part was wrong? This way maybe others who have gotten the summary can also make sure to note it :)
Par: rabuca • 2 année de cela
hey, I thought its a big thing but it's just something small: its about the interpretation of the F statistic, where it says F<1 means that MSR is greater than MSR
Par: galinajimberry • 2 année de cela
ah, alright! thank you for letting me know :)
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Week 3: General Linear ModelCh 12: Comparing Several Independent Means
Using a Linear Model to Compare Several Means
Advantages of the linear model framework:
1) The linear model extends logically to the more complex situations – e.g., multiple
predictors, unequal group sizes
2) SPSS uses the linear model framework (i.e., the general linear model) for comparing
means
The general equation is used again => 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑖 = (𝑚𝑜𝑑𝑒𝑙) + 𝑒𝑟𝑟𝑜𝑟𝑖
1. ‘Model’ in the equation => replaced with one dummy variable that codes two groups
(i.e., 0 for one group, and 1 for the other)
- And an associated b-value => representing the difference between the group
means
Any number of groups can be included => by extending the number of dummy variables to
one less than the number of groups
- That one less group => i.e., the baseline category; assigned a 0 code on all
dummy variables
The baseline category => the condition against which the other groups are being compared
• In most experiments => this would be the control group
• In designs with unequal group sizes => important that the baseline category contains
a large number of cases
- Ensures that the estimates of b-values are reliable
𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑖 = 𝑏0 + 𝑏1 𝑔𝑟𝑜𝑢𝑝1 + 𝑏2 𝑔𝑟𝑜𝑢𝑝2 + 𝜀𝑖
,Using a coding scheme => allows for each
group to be uniquely expressed by the
combined values for the dummy variables
When predicting an outcome from group membership => the predicted values from the model
are the group means
The colorized horizontal lines => represent the mean
outcome of each group
The shapes => represent the outcome of individual
participants (different shapes => different
experimental groups)
The red horizontal line => the average outcome of all
participants
The model for the control group – in which all dummies are coded 0 – becomes:
𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑖 = 𝑏0 + (𝑏1 × 0) + (𝑏2 × 0)
Therefore, the bo in the model => always the mean of the baseline category
For someone in group 1 – the value of the dummy variable for group 1 will be coded 1 and the
value for group 2 will be coded 0 – becoming:
𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑖 = 𝑏0 + (𝑏1 × 1) + (𝑏2 × 0)
This shows that the predicted outcome for someone in group 1 => the sum of bo and the b for
the dummy variable for group 1 (= b1)
Therefore, the mean of the control group = bo and the predicted outcome for someone in group
1 = the mean of that group
• We can replace bo and outcome with:
𝑋̅𝑔𝑟𝑜𝑢𝑝1 = 𝑋̅𝑐𝑜𝑛𝑡𝑟𝑜𝑙 + 𝑏1
𝑏1 = ̅𝑋𝑔𝑟𝑜𝑢𝑝1 − 𝑋̅𝑐𝑜𝑛𝑡𝑟𝑜𝑙
, • Shows that the b-value for the dummy variable representing group 1 => is the difference
between the means of that group and the control
Similarly => for group 2 the equation becomes:
𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑖 = 𝑏0 + (𝑏1 × 0) + (𝑏2 × 1)
𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑖 = 𝑏0 + 𝑏2
By replacing the variables => we can see that the b-value for group 2 represents the difference
between the means for that group and the control:
𝑋̅𝑔𝑟𝑜𝑢𝑝2 = 𝑋̅𝑐𝑜𝑛𝑡𝑟𝑜𝑙 + 𝑏2
𝑏2 = ̅𝑋𝑔𝑟𝑜𝑢𝑝2 − 𝑋̅𝑐𝑜𝑛𝑡𝑟𝑜𝑙
The output shows that the model fit was tested
with an F-statistic => which is significant
- The model represents the group means =>
this F tells us that using group means to
predict outcome scores is sig better than using
the mean of all scores
- i.e., the group means are sig different
The F-test is an overall test – and does not identify the differences between specific means
• The model parameters (b-values) do identify these differences
- The constant bo => the mean of the baseline group
- The b1 => the difference between the means of group 1 and the baseline group
- The b2 => the difference between the means of group 2 and the baseline group
Using the significance values of the associated t-tests => see that the difference between group
1 and control group (= b1) is significant at p = .008
- But the difference b/n group 2 and the control is not sig at p = .282
Logic of the F-Statistic
, The F-statistic => tests the overall fit of a linear model to a set of observed data
• It is the ratio of how good the model is – compared to how bad it is (error)
When the model is based on group means => the predictions from the model are those means
• If the group means are the same => ability to predict the observed data will be poor
(small F)
• If the means differ => able to better discriminate between cases from different groups
(large F)
In this context => F tells us whether the group means are significantly different
If the Ho: group means are equal => is true
- Then the b coefficients would all be 0 => because if the group means are equal
– then the difference between them will be 0
Mini Summary
• The model that represents ‘no effect’ or ‘no relationship b/n predictor variable and
outcome’ => one where the predicted value of the outcome is always the grand mean
- i.e., the mean of the outcome variable
• A different model can be fit to the data that represents the Ha
• The fit of the Ha model can be compared to the fit of the Ho model => i.e., using the
grand mean
• The intercept (bo) and one or more parameters (b) => describe the model
• The parameters determine the shape of the model fitted
- The bigger the coefficients => the greater the deviation b/n model and Ho model
(grand mean)
• In experimental research => the b parameters represent the differences b/n group means
- The bigger the differences b/n group means => the greater the difference b/n
model and the Ho model (grand mean)
• If the differences b/n group sizes are large enough => the resulting model will be a
better fit to the data than the Ho model (grand mean)
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