BIVARIATE DATA data for two variables

Transcription

1 (Chapter 3) BIVARIATE DATA data for two variables INVESTIGATING RELATIONSHIPS We have compared the distributions of the same variable for several groups, using double boxplots and back-to-back stemplots. Now, we will compare the distributions of several variables for the same group of individuals. We can investigate linear relationships between the variables and provide a model that can be used to predict values of a response variable using values of an explanatory varaible. To gain insight into the relationship between two quantitative variables, the following graphs can be used: Scatterplot Residual plot Scatterplot used to investigate the relationship between two quantitative variables, to show how much one variable is related to the other A scatterplot is used to display the strength, direction, and form of the relationship between two quantitative variables measured on the same group of individuals. When the variables are related, the x-axis contains the explanatory variable, the reason/explanation for the outcome, and the y-axis contains the response variable, the outcome/resulting response. When the scatterplot shows a linear relationship, we can interpret the scatterplot by looking for an overall pattern and for deviations from that pattern [outliers (in the y direction) and influential observations (in the x direction)] and by describing the strength, direction, and form of the relationship. Look for patterns: Strength: Pattern of close points with little scatter indicates strong relationship while scattered, spread-out points indicates weak relationship Direction: Increasing positive direction associates larger values of one variable with larger values of the other variable while decreasing negative direction associates larger values of one variable with smaller values of the other variable Form: Linear, nonlinear (curved/exponential, cluster, etc.) Indicators of categorical information can be added to the points on the scatterplot using symbols.

2 Residual plot graph of the residuals (on the vertical axis) and the independent variable (on the horizontal axis) A residual plot is a graph of the residuals versus the x-values, with a horizontal regression line at y = 0. It is used to verify that a linear model is appropriate for a data set. Evidence of a linear relationship appears as a graph of points with no pattern, randomly scattered above and below the horizontal line, without shape or direction and without curves. A definite pattern indicates that a nonlinear model is appropriate instead. A residual is the difference between an observed value (an actual y-value from the data set) and a predicted value (a predicted y-value from the linear regression equation) y - y ˆ. The mean of the residuals is always zero. Residuals on calculator To create a list of residuals in calculator: 1. Enter x-values in L 1 and y-values in 2. 2 nd, LIST, RESID, Enter 3. STO, 2 nd, L 3 To view residual plot on calculator: 1. 2 nd, STAT PLOT, Enter 2. Choose scatterplot (first graph top row) 3. Assign Xlist as x-values and Ylist as residuals, Enter 4. ZOOM, 9 To show residuals in calculator: 1. 2 nd Catalog, Diagnostics On, Enter 2. STAT, CALC, LinReg a + bx, Enter 3. 2 nd STAT PLOT, ZOOM, 9 L 2

3 Correlation When the scatterplot shows a fairly straight line and there is no pattern in the residual plot, correlation coefficient r may be a useful numerical summary/measurement of the strength of the linear relationship (but it is not a complete description of two-variable data). Correlation coefficient r measures the strength (and indicates the direction) of a linear relationship between any two quantitative variables. Correlation r values range from -1 to 1, with a strong correlation near these extremes and a weaker correlation at values near 0. Correlation is 0 if it does not yield a linear scatterplot, but not necessarily that there is no relationship; correlation is 1 or -1 when the linear relationship is exact, but correlation 1 or -1 does not guarantee a linear relationship. Don t trust your eyes to interpret the strength of a linear relationship on a scatterplot; consider the scaling, zoom in/out. Correlation coefficient r: r = 1 Σ n 1 (x i x s x ) ( y i y s y ) where: x i each x-value y i each y-value standard deviation of x-values s x (Calculator: STAT, CALC, 2-Var Stats) Although standard deviation and, therefore, slope, are affected by what units you choose for x and y, correlation r is not affected by what units you choose for x and y nor which variable is labeled x and y. When you reverse x and y [ (x, y) (y, x)], correlation r stays the same but the regression line will be different. If you want to make predictions in the other direction (the effect of y on x), then you must create another model completely. Calculated using means and standard deviations, the correlation coefficient is not resistant, so use it cautiously when the data set contains outliers. With standardized values, r is equal to the slope of the regression line. Remember that evidence of a relationship is not evidence of causation and that lurking variables may be affecting the variables, resulting in what may only appear to be a correlation. Calculator to compute correlation r: 1) Enter data values in L 1 and L 2 (Calculator: STAT, Edit) 2) Define L 3 : L 3 = ( L 1 x ) ( L 2 y ) (Calculator to find variables: VARS, Statistics) 1 s x s y 3) Enter formula: sum(l n 1 3) (Calculator to find sum : 2 nd, LIST, MATH)

4 Least Squares Regression A least-squares regression line (y = a + bx) is a mathematical model for the overall pattern of a linear relationship between an explanatory variable and a response variable. Using the scatterplot and patterns (and some formulas/calculator), we can create an equation to summarize the linear explanatory/response relationship between the two quantitative variables by performing least-squares regression. This best-fitting straight line, known as the Least Squares Regression Line, LSRL, is the line that minimizes the sum of squares of the vertical distances of data points from the line. This regression line describes how response variable y changes as explanatory variable x changes. It is created using means x and y, standard deviations s x and s y, and correlation r, and it always passes through the point (x, y ). The LSRL is used to predict y-values for x-values, but beware of extrapolation, predicting for x-values that are beyond the scope of the original data set. Least Squares Regression Line, LSRL: y = a + bx where: b = r s y (b: slope) s x a = y bx (a: y-intercept) (Calculator: STAT, CALC, LinReg(a+bx)) The LSRL equation in point-slope form: ŷ = predicted y-value æ y - y = r S ö y ç (x - x). è ø Be sure to interpret the slope in context of the problem and to express it as, For each additional (unit) in (variable x), the predicted (variable y) should (increase/decrease) by (slope, in units) on average. S x SAMPLE: yˆ x For each additional inch in length, the predicted weight should increase by 1.5 pounds on average. Although correlation r and a regression line can be found for any two variables, they are only useful for linear relationships.

5 Coefficient of determination While the variation in the residuals allows us to assess how well the regression model fits the data, the coefficient of determination r 2 allows us to assess the usefulness of the LSRL as it is the ratio of the variation in the values of y that is explained by the least-squares regression of y on x. While correlation r is given as a decimal (between -1 and +1), the coefficient of determination r 2 is usually given as a percentage (with 100% as a perfect fit). Remember to interpret r 2 in context of the problem. We will write a sentence in the form, About (%) of the variability in (variable y) is explained by the LSRL of (variable y) on (variable x). Computer Regression With computer outputs, we focus on y-intercept a, slope b, independent variable x, dependent variable y, correlation coefficient r, and coefficient of determination r 2. SAMPLE 1: SAMPLE 2:

6 EXAMPLE: a. Write the LSRL for the following computer print-out. b. Write a sentence to interpret coefficient of determination. EXAMPLE: a. Write the LSRL for the following computer print-out.

7 b. Write a sentence to interpret coefficient of determination. EXAMPLE: An article in the August 1997 issue of the Journal of the American Medical Association reported on a survey that tracked emergency room visits at randomly selected hospitals nationwide. The data on the number of jet skis in use, the number of accidents, and the number of fatalities for the years follow: Year Number in use Accidents Fatalities , , , ,376 1, ,915 1, ,283 1, ,545 2, ,000 3, ,000 4, ,000 4, Examine the relationship between number of jet skis in use and number of accidents. 1) Make a scatter plot (title, any necessary breaks, labeled axes including units, etc.) (Calculator) Enter data: STAT, EDIT, explanatory variable in L 1, response variable in L 2. (Calculator) View scatter plot: 2 nd, STAT PLOT, etc. Interpret the scatter plot. --Address the strength, direction, and form. There is a very strong positive linear relationship between the number of jet skis in use and the number of accidents.

8 2) View and interpret the residual plot, a scatterplot of x versus residuals, to determine if LSRL is a good fit, the suitability of LRSL. The residual plot shows no distinct pattern, so we will proceed with linear regression. 3) Calculate correlation coefficient r, and determine if value is reasonable. --Formula for correlation coefficient: r = 1 n -1 -Work for correlation coefficient: å æ ç è x i - x S x ö æ y i - y ö ç ø è S y ø (Calculator) Find mean and standard deviation of each variable: STAT, CALC, 2-Var stats. r = 1 æ 92, ,227ö æ 376-1,947ö æ 900, ,227ö æ 4,010-1,947ö ç ç ç ç 10-1è 273,923 ø è 1,335 ø è 273,923 ø è 1,335 ø -Value of correlation coefficient: (Calculator) Find correlation coefficient r: STAT, CALC, LinReg(a+bx) (Make sure calculator is programmed for DiagnosticOn under 2 nd, CATALOG.) r = Address reasonableness by noting strength and direction. The correlation coefficient is reasonable because it shows a very strong positive linear correlation between number of jet skis in use and number of accidents. 4) Find LSRL and plot. --Use point-slope form for Least Squares Regression Line (see formula in above notes). (show formula work here ) we can also sometimes use a and b for Least Squares Regression Line (see formula in above notes). (Calculator) Find a and b: STAT, CALC, LinReg(a+bx) y ˆ = a + bx y ˆ = x Number of accidents = (number of jet skis in use) --Plot the LSRL onto scatter plot by making up any two x-values that are not part of data set and plugging into LSRL to find y. Plot and connect to form a line. (150,000, ) and (340,000, ) (150,000, 719.2) and (340,000, ) (Label points using a symbol/color to differentiate between actual points.) 5) Interpret slope.

9 --Put in terms of unit change: as x goes up by 1, y (increases/decreases) by b. ( For each additional (unit) in (x variable), the predicted (y variable) should (increase/decrease) by (slope, in units) on average. ) For each additional number of jet skis in use, the predicted number of accidents should increase by.0048 on average. 6) Compute and interpret the coefficient of determination r 2, the proportion of variation that is defined by the line, the ratio of explained error to total error. --Calculate SST and SSE: å SST (Sum of Squares for Total) (y - y ) 2 (1) Highlight L 3 and define as (L 2 - y ) 2. (Find y under VARS, Statistics) (2) Find sum of L 3. (2 nd, STAT, MATH, sum) SST is 16,042, SSE (Sum of Squares for Error) å (y - y ˆ ) 2 (1) Highlight L 4 and define as (L 2 -Y 1 (L 1 )) 2. (Find Y 1 under VARS, Y-VARS, Function) (2) Find sum of L 4. (2 nd, STAT, MATH, sum) SSE is 283, Use SST and SSE to find r 2 : r 2 = SST - SSE SST = SSP SST The coefficient of determination is Interpretation: About 98% of the variability in the number of accidents is explained by the LSRL of the number of accidents on the number of jet skis in use. 7) Predict the number of accidents when the number of jet skis in use is 200, Input 200,000 as the x-variable into the LSRL to make the prediction. y ˆ = x y ˆ = (200,000) y ˆ = *Be careful to not extrapolate beyond the range of data points. 8) Show how the residual value for 600,000 jet skis in use is calculated. --Residuals (errors): vertical distances from the points to the line (Observed value) (Predicted value of y) y - y ˆ When x = 600,000, y = 3002 (found from table of data values)

10 Residual: y - ˆ y 3002 Y 1 (600,000) (Calculator) 2 nd, STAT, RESID, STO( L 5 or ). Scroll to same line as the given x in L 1 the residual on that same line (600,000) to find Positive residuals indicate LSRL is below the y-value (actual y is above predicted y), and negative residuals indicate the LSRL is above the y-value.