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Notes Bivariate Data Chapters 7 - 9

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Bivariate Data Explores relationships between two quantitative variables.

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The explanatory variable attempts to explain the observed outcomes. (In algebra this is your independent variable – “x”)

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The response variable measures an outcome of a study. (In algebra this is your dependent variable – “y”)

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When we gather data, we usually have in mind which variables are which. Beware! – this explanatory/response relationship suggests a cause and effect relationship that may not exist in all data sets. Use common sense!!

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Displaying the Variables We always graph our data right? You use a scatterplot to graph the relationship between 2 quantitative variables. Each point represents an individual.

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Remember that not all relationships are linear!!! We will talk about non-linear in the next unit.

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Interpret a Scatterplot Here is what we look for: 1) direction (positive, negative) D 2) form ( linear, or not linear) S 3) strength ( correlation, r) S 4) deviations from the pattern (outliers) U SUDS!!

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Remember on outlier is an individual observation that falls outside the overall pattern of the graph. There is no outlier test for bivariate data. It’s a judgment call

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Categorical variables can be added to scatterplots by changing the symbols in the plot. (See P. 199 for examples) Visual inspection is often not a good judge of how strong a linear relationship is. Changing the plotting scales or the amount of white space around a cloud of points can be deceptive. So….

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Facts about Correlation: 1) positive r – positive association (positive slope) negative r – negative association (negative slope) 2) r must fall between –1 and 1 inclusive. 3) r values close to –1 or 1 indicate that the points lie close to a straight line. 4) r values close to 0 indicate a weak linear relationship. 5) r values of –1 or 1 indicate a perfect linear relationship. 6) correlation only measures the strength in linear relationships (not curves). 7) correlation can be strongly affected by extreme values (outliers).

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Least-Squares Regression Line The least-squares regression line (LSRL) is a mathematical model for the data. This line is also known as the line of best fit or the regression line.

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Formal definition… The least-squares regression line of y on x is the line that makes the sum of the squares of the vertical distances of the data points from the line as small as possible.

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The form…

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Some new formulas…

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Why do we do regression? The purpose of regression is to determine a model that we can use for making predictions.

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Communication is always the goal!!! When we write the equation for a LSRL we do not use x & y, we use the variable names themselves… For example: Predicted score = 52 + 1.5(hours studied)

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Another measure of strength… The coefficient of determination, r 2, is the fraction of the variation in the value of y that is explained by the linear model. When we explain r 2 then we say… ___% of the variability in ___(y) can be explained by this linear model.

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Deviations for single points A residual is the vertical difference between an actual point and the LSRL at one specific value of x. That is, Residual = observed y – predicted y or Residual = y – The mean of the residuals is always zero.

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A new plot… A residual plot plots the residuals on the vertical axis against the explanatory variables on the horizontal axis. Such a plot magnifies the residuals and makes patterns easier to see.

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Why do I need a residual plot? Remember that all data is not linear in shape!!! The residual plot clearly shows if linear is appropriate. A residual plot show good linear fit when the points are randomly scattered about y = 0 with no obvious patterns.

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To create a residual plot on the calculator: 1)You must have done a linear regression with the data you wish to use. 2) From the Stat-Plot, Plot # menu choose scatterplot and leave the x list with the x values. 3) Change the y-list to “RESID” chosen from the list menu. 4) Zoom – 9

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In scatterplots we can have points that are outliers or influential points or both. An observation can be an outlier in the x direction, the y direction, or in both directions. An observation is influential if removing it or adding it) would markedly change the position of the regression line.

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Extrapolation is the use of a regression model for prediction outside the domain of values of the explanatory variable x. Such predictions cannot be trusted.

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Association vs. Causation A strong association between two variables is NOT enough to draw conclusions about cause & effect.

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Association vs Causation Strong association between two variables x and y can reflect: A) Causation – Change in x causes change in y B) Common response – Both x and y are Responding to some other unobserved factor C) Confounding – the effect on y of the explanatory variable x is hopelessly mixed up with the effects on y of other variables.

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A Lurking Variable is a variable that has an important effect on the relationship among the variables in a study but is not included among the variables being studied. Lurking variables can suggest a relationship when there isn’t one or can hide a relationship that exists.

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Association vs Causation Cause and Effect can only be determined from a well designed experiment.

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Data with no apparent linear relationship can also be examined in two ways to see if a relationship still exists: 1) Check to see if breaking the data down into subsets or groups makes a difference. 2) If the data is curved in some way and not linear, a relationship still exists. We will explore that in the next chapter.

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