Some students are taking harder courses, like chemistry or statistics; some are smarter; some study effectively; and some get lucky and find that the professor has asked them exactly what they understood best. For each level of amount studied, there will be a distribution of grades. If there is a relationship between studying and grades, the location of that distribution of grades will change in an orderly manner as you move from lower to higher levels of studying.
Regression analysis is one of the most used and most powerful multivariate statistical techniques for it infers the existence and form of a functional relationship in a population. Once you learn how to use regression, you will be able to estimate the parameters — the slope and intercept — of the function that links two or more variables. With that estimated function, you will be able to infer or forecast things like unit costs, interest rates, or sales over a wide range of conditions.
Though the simplest regression techniques seem limited in their applications, statisticians have developed a number of variations on regression that greatly expand the usefulness of the technique. In this chapter, the basics will be discussed. Once again, the t-distribution and F-distribution will be used to test hypotheses. Before starting to learn about regression, go back to algebra and review what a function is. More intuitively, if there is a regular relationship between two variables, there is usually a function that describes the relationship.
Functions are written in a number of forms. The simplest functional form is the linear function where:. There can be functions where one variable depends on the values values of two or more other variables where x 1 and x 2 together determine the value of y. There can also be non-linear functions, where the value of the dependent variable y in all of the examples we have used so far depends on the values of one or more other variables, but the values of the other variables are squared, or taken to some other power or root or multiplied together, before the value of the dependent variable is determined.
Regression allows you to estimate directly the parameters in linear functions only, though there are tricks that allow many non-linear functional forms to be estimated indirectly. Regression also allows you to test to see if there is a functional relationship between the variables, by testing the hypothesis that each of the slopes has a value of zero. First, let us consider the simple case of a two-variable function.
You believe that y , the dependent variable, is a linear function of x , the independent variable — y depends on x.
Collect a sample of x , y pairs, and plot them on a set of x , y axes. The basic idea behind regression is to find the equation of the straight line that comes as close as possible to as many of the points as possible.
The parameters of the line drawn through the sample are unbiased estimators of the parameters of the line that would come as close as possible to as many of the points as possible in the population, if the population had been gathered and plotted.
In keeping with the convention of using Greek letters for population values and Roman letters for sample values, the line drawn through a population is:.
In most cases, even if the whole population had been gathered, the regression line would not go through every point. Most of the phenomena that business researchers deal with are not perfectly deterministic, so no function will perfectly predict or explain every observation.
Imagine that you wanted to study the estimated price for a one-bedroom apartment in Nelson, BC. You decide to estimate the price as a function of its location in relation to downtown.
If you collected 12 sample pairs, you would find different apartments located within the same distance from downtown. In other words, you might draw a distribution of prices for apartments located at the same distance from downtown or away from downtown.
Because the best that can be expected is to predict the mean price for a certain location, researchers often write their regression models with an extra term, the error term , which notes that many of the members of the population of location, price of apartment pairs will not have exactly the predicted price because many of the points do not lie directly on the regression line. In estimating the unknown parameters of the population for the regression line, we need to apply a method by which the vertical distances between the yet-to-be estimated regression line and the observed values in our sample are minimized.
This minimized distance is called sample error, though it is more commonly referred to as residual and denoted by e. In more mathematical form, the difference between the y and its predicted value is the residual in each pair of observations for x and y. Obviously, some of these residuals will be positive above the estimated line and others will be negative below the line.
If we add all these residuals over the sample size and raise them to the power 2 in order to prevent the chance those positive and negative signs are cancelling each other out, we can write the following criterion for our minimization problem:.
S is the sum of squares of the residuals. By minimizing S over any given set of observations for x and y , we will get the following useful formula:. After computing the value of b from the above formula out of our sample data, and the means of the two series of data on x and y , one can simply recover the intercept of the estimated line using the following equation:. For the sample data, and given the estimated intercept and slope, for each observation we can define a residual as:.
Depending on the estimated values for intercept and slope, we can draw the estimated line along with all sample data in a y — x panel. Such graphs are known as scatter diagrams. Consider our analysis of the price of one-bedroom apartments in Nelson, BC. The graph shown in Figure 8. In order to plot such a scatter diagram, you can use many available statistical software packages including Excel, SAS, and Minitab. In this scatter diagram, a negative simple regression line has been shown.
The estimated equation for this scatter diagram from Excel is:. One might also be curious about the fitted values out of this estimated model. You can simply plug the actual value for x into the estimated line, and find the fitted values for the prices of the apartments.
The residuals for all 12 observations are shown in Figure 8. You should also notice that by minimizing errors, you have not eliminated them; rather, this method of least squares only guarantees the best fitted estimated regression line out of the sample data. In the presence of the remaining errors, one should be aware of the fact that there are still other factors that might not have been included in our regression model and are responsible for the fluctuations in the remaining errors.
By adding these excluded but relevant factors to the model, we probably expect the remaining error will show less meaningful fluctuations. In determining the price of these apartments, the missing factors may include age of the apartment, size, etc. Because this type of regression model does not include many relevant factors and assumes only a linear relationship, it is known as a simple linear regression model.
Understanding that there is a distribution of y apartment price values at each x distance is the key for understanding how regression results from a sample can be used to test the hypothesis that there is or is not a relationship between x and y. If another sample of the same size is taken, another sample equation could be generated. Because the standard deviation of this sampling distribution is seldom known, statisticians developed a method to estimate it from a single sample.
With this estimated s b , a t-statistic for each sample can be computed:. Computing s b is tedious, and is almost always left to a computer, especially when there is more than one explanatory variable.
The estimate is based on how much the sample points vary from the regression line. If the points in the sample are not very close to the sample regression line, it seems reasonable that the population points are also widely scattered around the population regression line and different samples could easily produce lines with quite varied slopes. Though there are other factors involved, in general when the points in the sample are farther from the regression line, s b is greater.
Rather than learn how to compute s b , it is more useful for you to learn how to find it on the regression results that you get from statistical software. It is often called the standard error and there is one for each independent variable. The printout in Figure 8. You will need these standard errors in order to test to see if y depends on x or not.
If the slope equals zero, then changes in x do not result in any change in y. Formally, for each independent variable, you will have a test of the hypotheses:. Substitute zero for b into the t-score equation, and if the t-score is small, b is close enough to zero to accept H a.
Figure 8. Remember to halve alpha when conducting a two-tail test like this. The degrees of freedom equal n — m -1, where n is the size of the sample and m is the number of independent x variables.
There is a separate hypothesis test for each independent variable. This means you test to see if y is a function of each x separately. By testing to see if the regression helps predict, you are testing to see if there is a functional relationship in the population. Imagine that you have found the mean price of the apartments in our sample, and for each apartment, you have made the simple prediction that price of apartment will be equal to the sample mean, y.
This is not a very sophisticated prediction technique, but remember that the sample mean is an unbiased estimator of population mean, so on average you will be right. For each apartment, you could compute your error by finding the difference between your prediction the sample mean, y and the actual price of an apartment. Now, you can make another prediction of how much each apartment in the sample may be worth by computing:.
Notice that the measures of these differences could be positive or negative numbers, but that error or improvement implies a positive distance. If you use the sample mean to predict the amount of the price of each apartment, your error is y — y for each apartment.
Squaring each error so that worries about signs are overcome, and then adding the squared errors together, gives you a measure of the total mistake you make if you want to predict y. To make this raw measure of the improvement meaningful, you need to compare it to one of the two measures of the total mistake. One compares the improvement to the mistakes still made with regression. The other compares the improvement to the mistakes that would be made if the mean was used to predict.
The second is called R 2 , or the coefficient of determination. All of these mistakes and improvements have names, and talking about them will be easier once you know those names. You should be able to see that:. In other words, the total variations in y can be partitioned into two sources: the explained variations and the unexplained variations. Further, we can rewrite the above equation as:.
Going back to the idea of goodness of fit, one should be able to easily calculate the percentage of each variation with respect to the total variations. In particular, the strength of the estimated regression model can now be measured. Since we are interested in the explained part of the variations by the estimated model, we simply divide both sides of the above equation by SST, and we get:.
Only in cases where an intercept is included in a simple regression model will the value of R 2 be bounded between zero and one. The closer R 2 is to one, the stronger the model is. Alternatively, R 2 is also found by:. This is the ratio of the improvement made using the regression to the mistakes made using the mean.
It would help a little bit if you knew that the predictors you have are uncorrelated from the well-known predictor whatever it is , or that well-known predictor is constant or nearly constant for your data: then at least you could say that something other than the well-known predictor does have an effect on the response.
The answer to the general question is that it depends of many factors with the main ones being 1 number of covariates 2 variance of the estimates and residuals. With a small sample you do not have much power to detect a difference from 0. So I would look at the estimated variance of the regression parameters. From my experience with regression 21 observations with 5 variables is not enough data to rule out variables.
So I would not be so quick to throw out variables nor get too enamored with the ones that appear significant. The best answer is to wait until you have a lot more data. Sometimes that is easy to say but difficult to do. I would look at stepwise regression, forward and backward regression just to see what variables get selected. If the covariates are highly correlated this may show very different sets of variables being selected. Bootstrap the model selection procedure as that will be revealing as to the sensitivity of variable selection to changes in the data.
You should calculate the correlation matrix for covariates. Maybe Frank Harrell will chime in on this. He is a real expert on variable selection. I think he would at least agree with me that you should not pick a final model based solely on these 21 data points.
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Create a free Team What is Teams? Learn more. Minimum number of observations for multiple linear regression Ask Question. Asked 9 years, 5 months ago. Active 6 months ago. Viewed 83k times. My aim is just finding the relation between variables Is my data set enough to do multiple regression?
My correlation matrix is as follow var 1 var 2 var 3 var 4 var 5 Y var 1 1. Intercept Improve this question. Add a comment. Active Oldest Votes. Both models are built using Regression Generalized glm. The model data output appears in the bottom window of the Model tab, as shown in the following image.
For details about the regression output, see Output From Linear Regression. This section describes the linear regression output. Note that output may vary slightly due to sampling. The significance codes indicate how certain we can be that the coefficient has an impact on the dependent variable. For example, a significance level of 0. Stated differently, we can be The significance codes shown by asterisks are intended for quickly ranking the significance of each variable.
Residual standard error: F-statistic: It is calculated on 9 Df for the coefficients and 50 Df for the residuals. The P-value associated with this F-value is very small 5. The P-value is a measure of how confident you can be that the independent variables reliably predict the dependent variable.
P stands for probability and is usually interpreted as the probability that test data does not represent accurately the population from which it is drawn. If the P-value is 0. For example, if the P-value were greater than 0.
Note that this is an overall significance test assessing whether the group of independent variables when used together reliably predict the dependent variable, and does not address the ability of any of the particular independent variables to predict the dependent variable. The ability of each individual independent variable to predict the dependent variable is addressed in the coefficients table.
See P-values for the regression coefficients. How Does Linear Regression Work? Y represents the price of each of the vintage wines observed in the auction. Independent Variable X. X is the time since vintage for each of the vintage wines observed in the auction. It is also referred to as a covariate. It is the slope of the linear line, for example, it shows how prices increase with the increase in the number of vintage years, or how wines become more expensive the longer they mature.
For other data sets, the trend can be the inverse, that is, the slope can be decreasing. Represents the unexplained variation in the target variable. It is treated as a random variable that picks up all the variation in Y that is not explained by X. Load the Wine data into RStat. Select Ident for the ID variable. Keep all the other variables as Input.
Click Execute to set up the Model Data. Select Regression as the Type of model. Select Linear as the Model Builder. This is the default option when Regression is selected. Click Execute to run the model. Summary of the Regression model built using lm. This is the title of the summary provided for the model.
It also specifies which R function has been used to build the model. The model in this case is built with the lm function. Summary of the Regression model built using lm : R Function Call. This section shows the call to R and the data set or subset used in the model. Min 1Q Median 3Q Max This shows the distribution of the residuals. The residuals are the difference between the prices in the training data set and the predicted prices by this model. A negative residual is an overestimate and a positive residual is an underestimate.
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