Notes Bivariate Data Chapters 7 - 9. Bivariate Data Explores relationships between two quantitative...
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Transcript of Notes Bivariate Data Chapters 7 - 9. Bivariate Data Explores relationships between two quantitative...
The explanatory variable attempts to explain the observed outcomes. (In algebra this is your independent variable – “x”)
The response variable measures an outcome of a study. (In algebra this is your dependent variable – “y”)
○ 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!!
○ 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.
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.
○Remember that not all bivariate relationships are linear!!! We will talk about non-linear in the next unit.
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!!
• 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
○ 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….
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).
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.
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.
Why do we do regression? ○ The purpose of regression is to
determine a model that we can use for making predictions.
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)
Another measure of strength…○ The coefficient of determination, r2,
is the fraction of the variation in the value of y that is explained by the linear model.
○ When we explain r2 then we say… ___% of the variability in ___(y) can be
explained by this linear model.
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 yor
Residual = y –
○ The mean of the residuals is always zero.
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.
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.
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
○ 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.
○ 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.
Association vs. Causation
○A strong association between two variables is NOT enough to draw conclusions about cause & effect.
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.
○ 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.