Initial Postings: Read and reflect on the assigned readings for the week. Then post what you thought was the most important concept(s), method(s), term(s), and/or any other thing that you felt was worthy of your understanding in each assigned textbook chapter.
Your initial post should be based upon the assigned reading for the week, so the textbook should be a source listed in your reference section and cited within the body of the text. Other sources are not required but feel free to use them if they aid in your discussion.
Also, provide a graduate-level response to each of the following questions:
- Causal inference is used as a secondary or tertiary tool in root cause analysis. Please explain how causal inference and root cause analysis are used in problem detection. Respond to this discussion board (DB) in the context of your field of employment. For example, if you are in I.T., respond to this DB by explaining the cause of a network failure; or if you are the in the food industry, use this DB to explain the cause of a recent decline in customer satisfaction.
[Your post must be substantive and demonstrate insight gained from the course material. Postings must be in the student’s own words – do not provide quotes!]
[Your initial post should be at least 250+ words and in APA format (including Times New Roman with font size 12 and double spaced)
Correlation vs Causality in Linear Regression Analysis
Chapter 6
© 2019 McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or distribution without the prior written consent of McGraw-Hill Education
Learning Objectives
Differentiate between correlation and causality in general and in the regression environment
Calculate partial and semi partial correlation
Execute inference for correlation regression analysis
Execute passive prediction using regression analysis
Execute inference for determining functions
Execute active prediction using regression analysis
Distinguish the relevance of model fit between active and passive prediction
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The Difference Between Correlation and Causality
Yi = fi(X1i, X2i, …, XKi) + Ui
We define as the determining function, since it comprises the part of the outcome that we can explicitly determine
Ui can only be inferred by solving Yi – fi(X1i, X2i, …, XKi)
Data-generating process as a framework for modeling causality
The reasoning established to measure an average treatment effect using sample means easily maps to this framework
Easily extends into modeling causality for multi-level treatments and multiple-treatments
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A causal relationship between two variables clearly implies co-movement.
If X casually impacts Y, then when X changes, we expect a change in Y
However, variables often move together even when there is no casual relationship between them
For example, height of two different children of ages 5 and 10. Since both the children are growing during these ages, their heights will generally move together. this co-movement is not due to causality – an increase in height by one child will not change in the height for the other.
The Difference Between Correlation and Causality
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Measurement of the co-movement between two variables in a dataset is captured through sample covariance or correlation:
Covariance: sCov(X,Y) =
Correlation: sCorr(X,Y) =
The Difference Between Correlation and Causality
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When there are more than two variables, e.g., Y, X1, X2, we can also measure partial correlation between two of the variables
Partial correlation between two variables is their correlation after holding one or more other variables fixed
The Difference Between Correlation and Causality
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Causality implies that a change in one variable or variables causes a change in another
Data analysis attempting to measure causality generally involves an attempt to measure the determining function within the data-generating process
Correlation implies that variables move together
Data analysis attempting to measure correlation is not concerned about the data-generating process and determining function, it uses standard statistical formulas (sample correlation, partial correlation) to assess how variables move together
The Difference Between Correlation and Causality
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The dataset is a cross-section of 230 grocery stores
AvgPrice = Average Price
AvgHHSize = Average Size of Households of Customers at that Grocery Store.
Regression Analysis for Correlation
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Sales = b + m1AvgPrice + m2AvgHHSize
Solving b, m1, m2:
Sales = 1591.54 – 181.66 × AvgPrice + 128.09 × AvgHHSize
This equation provides us information about how the variables in the equation are correlated within our sample.
Regression Analysis for Correlation
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Unconditional correlation is the standard measure of correlation between two variables X and Y
Corr(X,Y) =
Sx = Sample standard deviation for X and
SY = Sample standard deviation for Y
Partial correlation between X and Y is a measure of the relationship between these two variables, holding at least one other variable fixed
Semi-partial correlation between X and Y is a measure of the relationship between these two variables, holding at least one other variable fixed for only X or Y
Different Ways to Measure Correlation Between Two Variables
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For the general regression equation: Y = b + m1X1 + … +mKXK the solutions for m1 through mk when solving the sample moment equations are proportional to the partial and semi-partial correlation between Y and the respective Xs
Regression Analysis for Correlation
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Suppose we have the data for the entire population for our grocery store data, then, we have:
Sales = B + M1AvgPrice + M2AvgHHSize
Capital letters are used to indicate that these are the intercept and slopes for the population, rather than the sample
Solve for B, M1, and M2 by solving the sample moment equations using the entire population of data
Regression and Population Correlation
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Regression and Population Criteria
We do not have the data for the entire population, but for a sample dataset for the population whose regression line is:
Sales = b + m1AvgPrice + m2AvgHHSize
Solve for b, m1 and m2
The intercept and slope(s) of the regression equation describing a sample are estimators for the intercept and slope(s) of the corresponding regression equation describing the population.
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Consistent estimator is an estimator whose realized value gets close to its corresponding population parameter as the sample size gets large.
Regression and Population Correlation
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Regression Line for Full Population
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Regression Lines for Three Samples of Size 10
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Regression Lines for Three Samples of Size 30
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In order to conduct hypothesis testing or building confidence intervals for the population parameters of a regression equation, we need to know the distribution of the estimators
Each estimator becomes very close to its corresponding population parameters for a large sample
For a large sample, these estimators are normally distributed
Confidence Interval and Hypothesis Testing for the Population Parameters
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A large random sample implies that:
b~N(B,σB)
m1~N(M1,σm1)
mk~N(MK,σmk)
If we write each element in the population as:
Yi = B + M1X1i + … + MKXK + Ei
, where Ei is the residual, then Var(Y|X) is equal to Var(E|X)
Common assumption that this variance is constant across all values of X , so Var(Y|X) = Var(E|X) = Var(E) = σ2
This consistency of variance is called homoscedasticity
Confidence Interval and Hypothesis Testing for the Population Parameters
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Sales = 1591.54 – 181.66 × AvgPrice + 128.09 × AvgHHSize
If Store A has an average price of $0.50 higher than Store B, and Store A has an average household size that is 0.40 less than Store B, then:
= -181.66 × 0.50 + 128.09 × (-0.4) = -142
We predict Store A has 143 fewer sales than Store B
When using correlational regression analysis to make predictions, we must be considering a population that spans across time and we assume that the population regression equation best describes the future population
Prediction Using Regression
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Regression and Causation
Data-generating process of an outcome Y can be written as:
Yi = fi(X1i, X2i, …, XKi) + Ui
We assume the determining function can be written as:
fi(X1i, X2i, …, XKi) = α + β1X1i + β2X2i +… βKXKi
Combining these assumptions into a single assumption, the data-generating process can be written as:
Yi = α + β1X1i + β2X2i +… βKXKi + Ui
Error term represents unobserved factors that determine the outcome
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Regression and Causation
Yi = B + M1X1i + … +MKXK + Ei (Correlation model)
Yi = α + β1X1i + … βKXKi + Ui (Causality model)
Correlational model residuals (Ei) have a mean of zero and are uncorrelated with each of Xs. For this model, we simply plot all the data points in the population and write each observation in terms of equation that best describes these points.
For the causality model, the data-generating process is the process that actually generating the data we observe and determining function need not be the equation that best describe the data.
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CONSIDERING THESE DATA FOR Y, X, AND U ARE FOR THE ENTIRE POPULATION:
THESE DATA WERE GENERATED USING THE DATA- GENERATING PROCESS: Yi = 5 + 3.2Xi + Ui
MEANING WE HAVE A DETERMING FUNCTION : f(X) = 5 + 3.2X
The Difference Between the Correlation Model and the Causality Model: An Example
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Scatterplot, Regression Line, and Determining Function of X and Y
IN THIS FIGURE, WE PLOT Y AND X ALONG WITH THE DETERMING FUNCTION (BLUE LINE) AND THE POPULATION REGRESSION EQUATION (RED LINE).
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Regression and Causation
The correlation model describes the data best but need not coincide with the causal mechanism generating the data
The causality model provides the casual mechanism but need not describe the data best
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The Relevance of Model Fit for Passive and Active Prediction
Total sum of squares (TSS): The sum of the squared difference between each observation of Y and the average value of Yi
TSS = Yi – )2
Sum of squared residuals (SSRes): The sum of the squared residuals.
SSRes = i
R-squared: The fraction of the total variance in Y that can be attributed to variation in the Xs
R2 = 1 – SSRes/TSS
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The Relevance of Model Fit for Passive and Active Prediction
A high R-squared implies a good fit, meaning the points on the regression equation tend to be close to the actual Y values
R-squared for passive prediction (correlation) : Finding a high R-squared implies the prediction is close to reality
R-squared for active prediction (causality): R-squared is not a primary consideration when evaluating predictions
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