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A plot of the residuals vs, the fitted \(y\)-values (the \(y\)-coordinates of the points on the original scatter-plot).about 95% of the residuals are between \(-2s\) and \(2s\).about 68% of the residuals are between \(-s\) and \(s\).If the distribution of the residuals is roughly bell-shaped, then the 68-95 Rule says that: You can also perform some simple diagnostics using the plot() function: plot(WeightEff) The relationship may be thought of as linear, so using lmGC() for tasks like prediction does make sense, for this data. Indeed, the cloud of point seems to follow a line fairly well. The scatter-plot includes the regression line.
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The single best way to check this is to look at a scatter-plot, so lmGC() comes equipped with a option to produce one: just set the argument graph to TRUE, as follows: lmGC(mpg ~ wt, data = mtcars, graph = TRUE) # The interval says you can be about 95% confident that the actual efficiency of the car is somewhere between 14.9 and 27.6 miles per gallon.īut we are getting ahead of ourselves! In order to use linear models at all, we need to make sure that our data really do show a linear relationship. You can also get a prediction interval for the response variable if you use the level argument: predict(WeightEff,x=3,level=0.95) # Predict mpg is about 21.25, (By the way, it also differs for different values of the explanatory variable.) It’s a little bit bigger than the residual standard error \(s\), and as a give-or-take figure it’s a bit more reliable. The “give-or-take” figure here is known as the prediction standard error. We predict that a 3000 pound car (from the time of the Motor Trend study that produced this data) would have a fuel efficiency of 21.25 mpg, give or take about 3.095 mpg or so.
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# give or take 3.095 or so for chance variation.
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Then if you want to predict the fuel efficiency of a car that weights 3000 pounds, run this command: predict(WeightEff, x = 3) # Predict mpg is about 21.25, If you think that you might want to use predict(), you may first want to store the model in a variable, for example: WeightEff <- lmGC(mpg ~ wt, data = mtcars) Its arguments are the linear model that is created by lmGC(), and the value x of the explanatory variable. When the value of the explanatory variable for an individual is known and you wish to predict the value of the response variable for that individual, you can use the predict() function. since \(R^2\) is fairly high, it seems that weight is a fairly decent predictor of fuel efficiency. We see here that about 75% of the variation in the fuel efficiency of the cars in the data set is accounted for by the variation in their weights. As a rough rule of thumb, we figure that when we use the regression equation to predict the fuel efficiency of a car from its weight, that prediction is liable to be off by about \(s\) miles per gallon, or so. From the slope of -5.34, we see that for every thousand pound increase in the weight of a car, we predict a 5.34 mpg decrease in the fuel efficiency. We need to look at a scatter plot to be really sure about it (for plots, see below), but at this point the value \(r = -0.87\) indicates a fairly strong negative linear relationship between fuel efficiency and weight. The output to the console is rather minimal. If you are interested in studying the relationship between the fuel efficiency of a car ( mpg in the mtcars data frame, measured in miles per gallon) and its weight ( wt in mtcars, measured in thousands of pounds), then you can run: lmGC(mpg ~ wt, data = mtcars) # Like many R functions, lmGC() accepts formula-data input. The following commands may be helpful in this regard: data(mtcars) In this tutorial we will work with a couple of data sets: mtcars from the data package that comes with the basic R installation, fuel from the tigerstats package and RailTrail from the package mosaicData, so make sure you become familiar with them. The function (and some of the data) that we will use comes from the tigerstats package, so make sure that it is loaded: require(tigerstats) Also, in lm() the explanatory variable(s) can even be factors! It’s only a starter-tool: by the end of the course, or in later statistics courses, you will move on to R’s function lm(), which allows you to work with more than one explanatory variable and which provides additional useful information in its output. The function lmGC() is a starter-tool for simple linear regression, when you are studying the relationship between two numerical variables, one of which you consider to be an explanatory or predictor variable and the other of which you think of as the response.