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Applied Business Forecasting and Planning. Simple Linear Regression. Simple Regression.

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### Applied Business Forecasting and Planning

Simple Linear Regression

Simple Regression

- Simple regression analysis is a statistical tool That gives us the ability to estimate the mathematical relationship between a dependent variable (usually called y) and an independent variable (usually called x).
- The dependent variable is the variable for which we want to make a prediction.
- While various non-linear forms may be used, simple linear regression models are the most common.

Introduction

- The primary goal of quantitative analysis is to use current information about a phenomenon to predict its future behavior.
- Current information is usually in the form of a set of data.
- In a simple case, when the data form a set of pairs of numbers, we may interpret them as representing the observed values of an independent (or predictor ) variable X and a dependent ( or response) variable Y.

Introduction

- The goal of the analyst who studies the data is to find a functional relation between the response variable y and the predictor variable x.

Regression Function

- The statement that the relation between X and Y is statistical should be interpreted as providing the following guidelines:

1. Regard Y as a random variable.

2. For each X, take f (x) to be the expected value (i.e., mean value) of y.

3. Given that E (Y) denotes the expected value of Y, call the equation

the regression function.

Historical Origin of Regression

- Regression Analysis was first developed by Sir Francis Galton, who studied the relation between heights of sons and fathers.
- Heights of sons of both tall and short fathers appeared to “revert” or “regress” to the mean of the group.

Construction of Regression Models

- Selection of independent variables
- Since reality must be reduced to manageable proportions whenever we construct models, only a limited number of independent or predictor variables can or should be included in a regression model. Therefore a central problem is that of choosing the most important predictor variables.
- Functional form of regression relation
- Sometimes, relevant theory may indicate the appropriate functional form. More frequently, however, the functional form is not known in advance and must be decided once the data have been collected and analyzed.
- Scope of model
- In formulating a regression model, we usually need to restrict the coverage of model to some interval or region of values of the independent variables.

Uses of Regression Analysis

- Regression analysis serves Three major purposes.
- Description
- Control
- Prediction
- The several purposes of regression analysis frequently overlap in practice

Formal Statement of the Model

- General regression model
- 0, and 1 are parameters
- X is a known constant
- Deviations are independent N(o, 2)

Meaning of Regression Coefficients

- The values of the regression parameters 0, and 1 are not known.We estimate them from data.
- 1 indicates the change in the mean response per unit increase in X.

Regression Line

- If the scatter plot of our sample data suggests a linear relationship between two variables i.e.

we can summarize the relationship by drawing a straight line on the plot.

- Least squares method give us the “best” estimated line for our set of sample data.

Regression Line

- We will write an estimated regression line based on sample data as
- The method of least squares chooses the values for b0, and b1 to minimize the sum of squared errors

Estimation of Mean Response

- Fitted regression line can be used to estimate the mean value of y for a given value of x.
- Example
- The weekly advertising expenditure (x) and weekly sales (y) are presented in the following table.

Point Estimation of Mean Response

- From previous table we have:
- The least squares estimates of the regression coefficients are:

Point Estimation of Mean Response

- The estimated regression function is:
- This means that if the weekly advertising expenditure is increased by $1 we would expect the weekly sales to increase by $10.8.

Point Estimation of Mean Response

- Fitted values for the sample data are obtained by substituting the x value into the estimated regression function.
- For example if the advertising expenditure is $50, then the estimated Sales is:
- This is called the point estimate (forecast) of the mean response (sales).

Example:Retail sales and floor space

- It is customary in retail operations to asses the performance of stores partly in terms of their annual sales relative to their floor area (square feet). We might expect sales to increase linearly as stores get larger, with of course individual variation among stores of the same size. The regression model for a population of stores says that

SALES = 0 + 1 AREA +

Example:Retail sales and floor space

- The slope 1 is as usual a rate of change: it is the expected increase in annual sales associated with each additional square foot of floor space.
- The intercept 0 is needed to describe the line but has no statistical importance because no stores have area close to zero.
- Floor space does not completely determine sales. The term in the model accounts for difference among individual stores with the same floor space. A store’s location, for example, is important.

Residual

- The difference between the observed value yi and the corresponding fitted value.
- Residuals are highly useful for studying whether a given regression model is appropriate for the data at hand.

Estimation of the variance of the error terms, 2

- The variance 2of the error terms i in the regression model needs to be estimated for a variety of purposes.
- It gives an indication of the variability of the probability distributions of y.
- It is needed for making inference concerning regression function and the prediction of y.

Regression Standard Error

- To estimate we work with the variance and take the square root to obtain the standard deviation.
- For simple linear regression the estimate of 2 is the average squared residual.
- To estimate , use
- s estimates the standard deviation of the error term in the statistical model for simple linear regression.

Basic Assumptions of a Regression Model

- A regression model is based on the following assumptions:
- There is a probability distribution of Y for each level of X.
- Given that µy is the mean value of Y, the standard form of the model is

where is a random variable with a normal distribution with mean 0 and standard deviation .

Conditions for Regression Inference

- You can fit a least-squares line to any set of explanatory-response data when both variables are quantitative.
- If the scatter plot doesn’t show an approximately linear pattern, the fitted line may be almost useless.

Conditions for Regression Inference

- The simple linear regression model, which is the basis for inference, imposes several conditions.
- We should verify these conditions before proceeding with inference.
- The conditions concern the population, but we can observe only our sample.

Conditions for Regression Inference

- In doing Inference, we assume:
- The sample is an SRS from the population.
- There is a linear relationship in the population.
- We can not observe the population , so we check the scatter plot of the sample data.
- The standard deviation of the responses about the population line is the same for all values of the explanatory variable.
- The spread of observations above and below the least-squares line should be roughly uniform as x varies.

Conditions for Regression Inference

- Plotting the residuals against the explanatory variable is helpful in checking these conditions because a residual plot magnifies patterns.

Analysis of Residual

- To examine whether the regression model is appropriate for the data being analyzed, we can check the residual plots.
- Residual plots are:
- Plot a histogram of the residuals
- Plot residuals against the fitted values.
- Plot residuals against the independent variable.
- Plot residuals over time if the data are chronological.

Analysis of Residual

- A histogram of the residuals provides a check on the normality assumption. A Normal quantile plot of the residuals can also be used to check the Normality assumptions.
- Regression Inference is robust against moderate lack of Normality. On the other hand, outliers and influential observations can invalidate the results of inference for regression
- Plot of residuals against fitted values or the independent variable can be used to check the assumption of constant variance and the aptness of the model.

Analysis of Residual

- Plot of residuals against time provides a check on the independence of the error terms assumption.
- Assumption of independence is the most critical one.

The residuals should have no systematic pattern.

The residual plot to right shows a scatter of the points with no individual observations or systematic change as x increases.

Residual plotsThe points in this residual plot have a curve pattern, so a straight line fits poorlyResidual plots

The points in this plot show more spread for larger values of the explanatory variable x, so prediction will be less accurate when x is large.Residual plots

Variable transformations

- If the residual plot suggests that the variance is not constant, a transformation can be used to stabilize the variance.
- If the residual plot suggests a non linear relationship between x and y, a transformation may reduce it to one that is approximately linear.
- Common linearizing transformations are:
- Variance stabilizing transformations are:

Inference about the Regression Model

- When a scatter plot shows a linear relationship between a quantitative explanatory variable x and a quantitative response variable y, we can use the least square line fitted to the data to predict y for a give value of x.
- Now we want to do tests and confidence intervals in this setting.

Inference about the Regression Model

- We think of the least square line we calculated from a sample as an estimate of a regression line for the population.
- Just as the sample mean is an estimate of the population mean µ.

Inference about the Regression Model

- We will write the population regression line as
- The numbers and are parameters that describe the population.
- We will write the least-squares line fitted to sample data as
- This notation reminds us that the intercept b0 of the fitted line estimates the intercept 0 of the population line, and the slope b1 estimates the slope 1 .

Confidence Intervals and Significance Tests

- In our previous lectures we presented confidence intervals and significance tests for means and differences in means.In each case, inference rested on the standard error s of the estimates and on t or z distributions.
- Inference for the slope and intercept in linear regression is similar in principal, although the recipes are more complicated.
- All confidence intervals, for example , have the form
- estimate t* Seestimate
- t* is a critical value of a t distribution.

Confidence Intervals and Significance Tests

- Confidence intervals and tests for the slope and intercept are based on the sampling distributions of the estimates b1 and b0.
- Here are the facts:
- If the simple linear regression model is true, each of b0 and b1 has a Normal distribution.
- The mean of b0 is 0 and the mean of b1 is 1.
- That is, the intercept and slope of the fitted line are unbiased estimators of the intercept and slope of the population regression line.

Confidence Intervals and Significance Tests

- The standard deviations of b0 and b1 are multiples of the model standard deviation .

Example:Weekly Advertising Expenditure

- Let us return to the Weekly advertising expenditure and weekly sales example. Management is interested in testing whether or not there is a linear association between advertising expenditure and weekly sales, using regression model. Use = .05

Example:Weekly Advertising Expenditure

- Test statistic:

Example:Weekly Advertising Expenditure

- Conclusion:

Since t =4.5 > 2.306 then we reject H0.

There is a linear association between advertising expenditure and weekly sales.

Confidence interval for 1

- Now that our test showed that there is a linear association between advertising expenditure and weekly sales, the management wishes an estimate of 1 with a 95% confidence coefficient.

Confidence interval for 1

- For a 95 percent confidence coefficient, we require t (.025; 8). From table B in appendix III, we find t(.025; 8) = 2.306.
- The 95% confidence interval is:

Example: Do wages rise with experience?

- Many factors affect the wages of workers: the industry they work in, their type of job, their education and their experience, and changes in general levels of wages. We will look at a sample of 59 married women who hold customer service jobs in Indiana banks. The following table gives their weekly wages at a specific point in time also their length of service with their employer, in month. The size of the place of work is recorded simply as “large” (100 or more workers) or “small.” Because industry, job type, and the time of measurement are the same for all 59 subjects, we expect to see a clear relationship between wages and length of service.

Example: Do wages rise with experience?

- From previous table we have:
- The least squares estimates of the regression coefficients are:

Example: Do wages rise with experience?

- What is the least-squares regression line for predicting Wages from Los?
- Suppose a woman has been with her bank for 125 months. What do you predict she will earn?
- If her actual wages are $433, then what is her residual?
- The sum of squared residuals for the entire sample is

Example: Do wages rise with experience?

- Do wages rise with experience?
- The hypotheses are:

H0: 1 = 0, Ha: 1 > 0

- The test statistics
- The P- value is:
- Conclusion:

Example: Do wages rise with experience?

- A 95% confidence interval for the average increase in wages per month of stay for the regression line in the population of all married female customer service workers in Indiana bank is
- The t distribution for this problem has n-2 = 57 degrees of freedom

Example: Do wages rise with experience?

- Regression calculations in Practice are always done by software.
- The computer out put for the case study is given in the following slide.

Using the regression Line

- One of the most common reasons to fit a line to data is to predict the response to a particular value of the explanatory variable.
- In our example, the least square line for predicting the weekly earnings for female bank customer service workers from their length of service is

Using the regression Line

- For a length of service of 125 months, our least-squares regression equation gives
- There are two different uses of this prediction.
- We can estimate the mean earnings of all workers in the subpopulation of workers with 125 months on the job.
- We can predict the earnings of one individual worker with 125 months of service.

Using the regression Line

- For each use, the actual prediction is the same, .But the margin of error is different for the two cases.
- To estimate the mean response, we use a confidence interval.
- To estimate an individual response y, we use prediction interval.
- A prediction interval estimates a single random response y rather than a parameter like µy

Using the regression Line

- The main distinction is that it is harder to predict for an individual than for the mean of a population of individuals.
- Each interval has the usual form
- The margin of error for the prediction interval is wider than the margin of error for the confidence interval.

Using the regression Line

- The standard error for estimating the mean response when the explanatory variable x takes the value x* is:

Using the regression Line

- The standard error for predicting an individual response when the explanatory variable x takes the value x* is:

Prediction of a new response ( )

- We now consider the prediction of a new observation y corresponding to a given level x of the independent variable.
- In our advertising expenditure and weekly sales, the management wishes to predict the weekly sales corresponding to the advertising expenditure of x = $50.

Interval Estimation of a new response ( )

- The following formula gives us the point estimator (forecast) for y.
- 1- % prediction interval for a new observation is:
- Where

Example

- In our advertising expenditure and weekly sales, the management wishes to predict the weekly sales if the advertising expenditure is $50 with a 90 % prediction interval.
- We require t(.05; 8) = 1.860

Example

- The 90% prediction interval is:

Analysis of variance approach to Regression analysis

- Analysis of Variance is the term for statistical analyses that break down the variation in data into separate pieces that correspond to different sources of variation.
- It is based on the partitioning of sums of squares and degrees of freedom associated with the response variable.
- In the regression setting, the observed variation in the responses (yi) comes from two sources.

Analysis of variance approach to Regression analysis

- Consider the weekly advertising expenditure and the weekly sales example. There is variation in the amount ($) of weekly sales, as in all statistical data. The variation of the yi is conventionally measured in terms of the deviations:

Analysis of variance approach to Regression analysis

- The measure of total variation, denoted by SST, is the sum of the squared deviations:
- If SST = 0, all observations are the same(No variability).
- The greater is SST, the greater is the variation among the y values.
- When we use the regression model, the measure of variation is that of the y observations variability around the fitted line:

Analysis of variance approach to Regression analysis

- The measure of variation in the data around the fitted regression line is the sum of squared deviations (error), denoted SSE:
- For our Weekly expenditure example

SSE = 36124.76

SST = 128552.5

- What accounts for the substantial difference between these two sums of squares?

Analysis of variance approach to Regression analysis

- The difference is another sum of squares:
- SSR stands for regression sum of squares.
- SSR is the variation among the predicted responses . The predicted responses lie on the least-square line. They show how y moves in response to x.
- The larger is SSR relative to SST, the greater is the role of regression line in explaining the total variability in y observations.

Analysis of variance approach to Regression analysis

- In our example:
- This indicates that most of variability in weekly sales can be explained by the relation between the weekly advertising expenditure and the weekly sales.

Formal Development of the Partitioning

- We can decompose the total variability in the observations yi as follows:
- The total deviation can be viewed as the sum of two components:
- The deviation of the fitted value around the mean .
- The deviation of yi around the fitted regression line.

Formal Development of the Partitioning

- Skipping quite a bit of messy algebra, we just state that this analysis of variance equation always holds:
- Breakdown of degree of freedom:

Mean squares

- A sum of squares divided by its degrees of freedom is called a mean square (MS)
- Regression mean square (MSR)
- Error mean square (MSE)
- Note: mean squares are not additive.

Mean squares

- In our example:

Analysis of Variance Table

- The breakdowns of the total sum of squares and associated degrees of freedom are displayed in a table called analysis of variance table (ANOVA table)

Analysis of Variance Table

- In our weekly advertising expenditure and weekly sales example the ANOVA table is:

Analysis of Variance Table

- The Analysis of Variance table reports in a different way quantities such as r2 and s that are needed in regression analysis.
- It also reports in a different way the test for the overall significance of the regression.
- If regression on x has no value for predicting y, we expect the slope of the population regression line to be close to 0.

Analysis of Variance Table

- That is the null hypothesis of “no linear relationship” is:
- We standardize the slope of the least-squares line to get a t statistic.

F-Test for 1= 0 versus 1 0

- The analysis of variance approach starts with sums of squares.
- If regression on x has no value for predicting y, we expect the SSR to be only a small part of the SST, most of which will be made of the SSE.
- The proper way to standardize this comparison is to use the ratio

F-Test for 1= 0 versus 1 0

- In order to be able to construct a statistical decision rule, we need to know the distribution of our test statistic F.
- When H0 is true, our test statistic, F, follows the F- distribution with 1, and n-2 degrees of freedom.
- Table C-5 on page 513 of your text gives the critical values of the F-distribution at = 0.05 and .01.

F-Test for 1= 0 versus 1 0

- Construction of decision rule:
- At = 5% level
- Reject H0 if
- Large values of F support Ha and Values of F near 1 support H0.

F-Test for 1= 0 versus 1 0

- Using our example again, let us repeat the earlier test on 1. This time we will use the F-test. The null and alternative hypothesis are:
- Let = .05. Since n=10, we require F(.05; 1, 8). From table 5-3 we find that F(.05; 1, 8) = 5.32. Therefore the decision rule is:
- Reject H0 if:

F-Test for 1= 0 versus 1 0

- From ANOVA table we have

MSR = 92427.74

MSE = 4515.6

- Our test statistic F is:
- Decision:
- Since 20.47> 5.32, we reject H0, that is there is a linear association between weekly advertising expenditure and weekly sales.

F-Test for 1= 0 versus 1 0

- Equivalence of F Test and t Test:
- For given level, the F test of 1 = 0 versus

1 0 is equivalent algebraically to the two sided t-test.

- Thus, at a given level, we can use either the t-test or the F-test for testing 1 = 0 versus

1 0.

- The t-test is more flexible since it can be used for one sided test as well.

Analysis of Variance Table

- The complete ANOVA table for our example is:

Computer Output

- The EXCEL out put for our example is:

Coefficient of Determination

- Recall that SST measures the total variations in yi when no account of the independent variable x is taken.
- SSE measures the variation in the yi when a regression model with the independent variable x is used.
- A natural measure of the effect of x in reducing the variation in y can be defined as:

Coefficient of Determination

- R2 is called the coefficient of determination.
- 0 SSE SST, it follows that:
- We may interpret R2 as the proportionate reduction of total variability in y associated with the use of the independent variable x.
- The larger is R2, the more is the total variation of y reduced by including the variable x in the model.

Coefficient of Determination

- If all the observations fall on the fitted regression line, SSE = 0 and R2 = 1.
- If the slope of the fitted regression line

b1 = 0 so that , SSE=SST and R2 = 0.

- The closer R2 is to 1, the greater is said to be the degree of linear association between x and y.
- The square root of R2 is called the coefficient of correlation.

Correlation Coefficient

- Recall thatthe algebraic expression for the correlation coefficient is.

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