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Multivariable Thinking & Sources of Variation

Examine variation in an outcome (response variable)

Sources of explained variation: How much can be explained by relationships with predictor (explanatory) variables
Sources of unexplained variation: variation in the response variable still left unexplained

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Multivariable Thinking & Sources of Variation

We explore variation in the response variable both graphically and numerically

Think about different sources of variation in the response variable
How does the variation in the response look both graphically & numerically when accounting for these different sources of variation?

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Housing Prices in Michigan

Students look at a sample of 13 homes that are for sale in an area north of Lake Macatawa in Michigan, USA.
How would you explore these prices?
Students use histograms and descriptive statistics to explore the quantitative variable of home price.

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Housing Prices in Michigan

Graph and statistics are from JMP

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Housing Prices in Michigan

Suppose a new home in the area goes up for sale, inquire what home price the students would predict? Mean? Median?
If we use the mean, we over-predict or under-predict many of the 13 homes already in our data set!
The difference between the actual home price and the predicted home price is called the residual.

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Housing Prices in Michigan

Dot plot of residuals using the mean housing price as the prediction for each house.
The SD (standard deviation) is a typical distance of home price from the mean

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Housing Prices in Michigan

Might there be more information we could collect on the houses that might explain the variability in price?
Use a Sources of Variation Diagram

Observed variation in:	Sources of explained variation	Sources of unexplained variation
Inclusion criteria:

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Housing Prices in Michigan

Might there be more information we could collect on the houses that might explain the variability in price?
Sources of Variation Diagram filled in

Observed variation in: Price	Sources of explained variation	Sources of unexplained variation
Inclusion criteria: Houses in Michigan north of Lake Macatawa		Size Number of stories Number of bedrooms Garage Location Other

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Housing Prices in Michigan

We first have students look at size of house as measured in square feet.

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Housing Prices in Michigan

How much of the variation in home price are we able to explain by size of home (square feet)?

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Housing Prices in Michigan

Dotplot of residuals using the regression line to predict the price for each house.
Notice the SD is a lot lower than when the mean was used as the predicted price (240,923 compared to 177,194)

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Housing Prices in Michigan

If the students compare the variances (SD²) of the residuals using the mean as their predicted home price to the variance (SD²) of the residuals using the regression line to predict the home price they find the variability of the residuals have roughly been cut in half: (177,194)²/(240,923)² = 0.54
This means 54% of the variability in home prices is still unexplained, but that square footage has explained 46% of the variability in home prices. (R²)

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Housing Prices in Michigan

How does our sources of variation diagram change?

Observed variation in: Price	Sources of explained variation	Sources of unexplained variation
Inclusion criteria: Houses in Michigan north of Lake Macatawa	Size	Number of stories Number of bedrooms Garage Location Other

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Housing Prices in Michigan

Might adding another variable to the analysis explain still more of the variation price that is left to explain? What about location?

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Housing Prices in Michigan

Ask students if adding the third variable of location has accounted for more of the variability in home prices. YES!
Residuals from the model that includes both square footage and lake front information for each house

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Housing Prices in Michigan

Now when compare the ratio of the variation still unexplained to the total variation, what can we say?
(42,850)²/(240,923)² = 0.03
3% of the variability in home prices is still unexplained, which means that square footage and lake location explain roughly 97% of the variability in home prices in this area.

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Housing Prices in Michigan

How does our sources of variation diagram change?

Observed variation in: Price	Sources of explained variation	Sources of unexplained variation
Inclusion criteria: Houses in Michigan north of Lake Macatawa	Size Location	Number of stories Number of bedrooms Garage Other

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Wisdom Gained

It is hoped that students now think about data in terms of explaining variability in the response variable by exploring relationships with possible explanatory variables
Exploring how multiple explanatory variables may work together to better explain the variability in the response variable

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Practice filling out a SOV diagram

Recall one our research studies from yesterday, or from today.
Fill out the SOV (Sources Of Variation) diagram

This helps us think about what is explaining the variability in our response
It also reminds us of the study design with inclusion criteria

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