Which regression coefficients pass the t test

Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Understanding t-test for linear regression Ask Question. Asked 3 years, 6 months ago. Active 1 year, 6 months ago. Viewed 61k times. So, when using a t-test on a linear regression, what is it that we're actually doing? Improve this question.

Add a comment. Active Oldest Votes. Improve this answer. Michael Hardy 6, 1 1 gold badge 20 20 silver badges 36 36 bronze badges. Thank you so much. But here is a much shorter nonmathematical version! With those values t and df you can determine the P value with an online calculator or table.

Harvey Motulsky Harvey Motulsky Rohan Moore Rohan Moore 1. Sign up or log in Sign up using Google. In an extreme case where is a linear combination of , correlation equal to one, both variables move in identical ways with Y. In this case it is impossible to determine the variable that is the true cause of the effect on Y. If the two variables were actually perfectly correlated, then mathematically no regression results could actually be calculated.

The normal equations for the coefficients show the effects of multicollinearity on the coefficients. The correlation between and , , appears in the denominator of both the estimating formula for and. If the assumption of independence holds, then this term is zero. This indicates that there is no effect of the correlation on the coefficient. On the other hand, as the correlation between the two independent variables increases the denominator decreases, and thus the estimate of the coefficient increases.

The correlation has the same effect on both of the coefficients of these two variables. This results in biased estimates. Multicollinearity has a further deleterious impact on the OLS estimates. The correlation between the two independent variables also shows up in the formulas for the estimate of the variance for the coefficients.

Here again we see the correlation between and in the denominator of the estimates of the variance for the coefficients for both variables. If the correlation is zero as assumed in the regression model, then the formula collapses to the familiar ratio of the variance of the errors to the variance of the relevant independent variable. If however the two independent variables are correlated, then the variance of the estimate of the coefficient increases.

This results in a smaller t-value for the test of hypothesis of the coefficient. In short, multicollinearity results in failing to reject the null hypothesis that the X variable has no impact on Y when in fact X does have a statistically significant impact on Y.

Said another way, the large standard errors of the estimated coefficient created by multicollinearity suggest statistical insignificance even when the hypothesized relationship is strong. In the last section we concerned ourselves with testing the hypothesis that the dependent variable did indeed depend upon the hypothesized independent variable or variables. It may be that we find an independent variable that has some effect on the dependent variable, but it may not be the only one, and it may not even be the most important one.

Remember that the error term was placed in the model to capture the effects of any missing independent variables. The multiple correlation coefficient , also called the coefficient of multiple determination or the coefficient of determination , is given by the formula:.

Figure shows how the total deviation of the dependent variable, y, is partitioned into these two pieces. Figure shows the estimated regression line and a single observation, x 1. Regression analysis tries to explain the variation of the data about the mean value of the dependent variable, y. The question is, why do the observations of y vary from the average level of y?

The value of y at observation x 1 varies from the mean of y by the difference. The sum of these differences squared is SST, the sum of squares total. We recall that this is the error term, e, and the sum of these errors is SSE, sum of squared errors. Sometimes the SSR is called SSM for sum of squares mean because it measures the deviation from the mean value of the dependent variable, y, as shown on the graph.

For time series studies expect a high R 2 and for cross-section data expect low R 2. While a high R 2 is desirable, remember that it is the tests of the hypothesis concerning the existence of a relationship between a set of independent variables and a particular dependent variable that was the motivating factor in using the regression model. It is validating a cause and effect relationship developed by some theory that is the true reason that we chose the regression analysis.

Increasing the number of independent variables will have the effect of increasing R 2. To account for this effect the proper measure of the coefficient of determination is the , adjusted for degrees of freedom, to keep down mindless addition of independent variables. There is no statistical test for the R 2 and thus little can be said about the model using R 2 with our characteristic confidence level. Two models that have the same size of SSE, that is sum of squared errors, may have very different R 2 if the competing models have different SST, total sum of squared deviations.

The goodness of fit of the two models is the same; they both have the same sum of squares unexplained, errors squared, but because of the larger total sum of squares on one of the models the R 2 differs. Again, the real value of regression as a tool is to examine hypotheses developed from a model that predicts certain relationships among the variables. These are tests of hypotheses on the coefficients of the model and not a game of maximizing R 2. Another way to test the general quality of the overall model is to test the coefficients as a group rather than independently.

Because this is multiple regression more than one X , we use the F-test to determine if our coefficients collectively affect Y. The hypothesis is:. If the null hypothesis cannot be rejected, then we conclude that none of the independent variables contribute to explaining the variation in Y. Reviewing Figure we see that SSR, the explained sum of squares, is a measure of just how much of the variation in Y is explained by all the variables in the model.

SSE, the sum of the errors squared, measures just how much is unexplained. It follows that the ratio of these two can provide us with a statistical test of the model as a whole. Remembering that the F distribution is a ratio of Chi squared distributions and that variances are distributed according to Chi Squared, and the sum of squared errors and the sum of squares are both variances, we have the test statistic for this hypothesis as:.

It can be shown that this is equivalent to:. As with all our tests of hypothesis, we reach a conclusion by comparing the calculated F statistic with the critical value given our desired level of confidence.

If the calculated test statistic, an F statistic in this case, is in the tail of the distribution, then we cannot accept the null hypothesis. By not being able to accept the null hypotheses we conclude that this specification of this model has validity, because at least one of the estimated coefficients is significantly different from zero. An alternative way to reach this conclusion is to use the p-value comparison rule. The p-value is the area in the tail, given the calculated F statistic.

In essence, the computer is finding the F value in the table for us. How to read the output of an Excel regression is presented below. This is the probability of NOT accepting a false null hypothesis. If this probability is less than our pre-determined alpha error, then the conclusion is that we cannot accept the null hypothesis. Thus far the analysis of the OLS regression technique assumed that the independent variables in the models tested were continuous random variables.

There are, however, no restrictions in the regression model against independent variables that are binary. This opens the regression model for testing hypotheses concerning categorical variables such as gender, race, region of the country, before a certain data, after a certain date and innumerable others. These categorical variables take on only two values, 1 and 0, success or failure, from the binomial probability distribution.

The form of the equation becomes:. X 2 is the dummy variable and X 1 is some continuous random variable. The constant, b 0 , is the y-intercept, the value where the line crosses the y-axis. In effect the dummy variable causes the estimated line to shift either up or down by the size of the effect of the characteristic captured by the dummy variable.

Note that this is a simple parallel shift and does not affect the impact of the other independent variable; X 1. This variable is a continuous random variable and predicts different values of y at different values of X 1 holding constant the condition of the dummy variable.

An example of the use of a dummy variable is the work estimating the impact of gender on salaries. There is a full body of literature on this topic and dummy variables are used extensively. For this example the salaries of elementary and secondary school teachers for a particular state is examined. Using a homogeneous job category, school teachers, and for a single state reduces many of the variations that naturally effect salaries such as differential physical risk, cost of living in a particular state, and other working conditions.

The estimating equation in its simplest form specifies salary as a function of various teacher characteristic that economic theory would suggest could affect salary. The results of the regression analysis using data on 24, school teachers are presented below.

The coefficients for all the independent variables are significantly different from zero as indicated by the standard errors. Dividing the standard errors of each coefficient results in a t-value greater than 1. The binary variable, our dummy variable of interest in this analysis, is gender where male is given a value of 1 and female given a value of 0. The coefficient is significantly different from zero with a dramatic t-statistic of 47 standard deviations. We thus cannot accept the null hypothesis that the coefficient is equal to zero.

Therefore we conclude that there is a premium paid male teachers of? It is important to note that these data are from some time ago and the? A graph of this example of dummy variables is presented below.

In two dimensions, salary is the dependent variable on the vertical axis and total years of experience was chosen for the continuous independent variable on horizontal axis. Any of the other independent variables could have been chosen to illustrate the effect of the dummy variable. The relationship between total years of experience has a slope of? If the gender variable is equal to 1, for male, the coefficient for the gender variable is added to the intercept and thus the relationship between total years of experience and salary is shifted upward parallel as indicated on the graph.

Also marked on the graph are various points for reference. A female school teacher with 10 years of experience receives a salary of? A more complex interaction between a dummy variable and the dependent variable can also be estimated. A confidence interval represents a closed interval where a certain percentage of the population is likely to lie.

This section discusses confidence intervals used in simple linear regression analysis. For the data in the preceding table , assume that a new value of the yield is observed after the regression model is fit to the data. This new observation is independent of the observations used to obtain the regression model. The prediction interval values calculated in this example are shown in the figure below as Low Prediction Interval and High Prediction Interval, respectively.

It is important to analyze the regression model before inferences based on the model are undertaken. The following sections present some techniques that can be used to check the appropriateness of the model for the given data. These techniques help to determine if any of the model assumptions have been violated. The coefficient of determination is a measure of the amount of variability in the data accounted for by the regression model.

The coefficient of determination is the ratio of the regression sum of squares to the total sum of squares. These values measure different aspects of the adequacy of the regression model.

The values of S, R-sq and R-sq adj indicate how well the model fits the observed data. Plots of residuals are used to check for the following:. Examples of residual plots are shown in the following figure. Such a plot indicates an appropriate regression model. Such a plot indicates increase in variance of residuals and the assumption of constant variance is violated here. If the residuals follow the pattern of c or d , then this is an indication that the linear regression model is not adequate.

A plot of residuals may also show a pattern as seen in e , indicating that the residuals increase or decrease as the run order sequence or time progresses. This may be due to factors such as operator-learning or instrument-creep and should be investigated further. Residual plots for the data of the preceding table are shown in the following figures. One of the following figures is the normal probability plot. It can be observed that the residuals follow the normal distribution and the assumption of normality is valid here.

Both of these plots show that the 21st observation seems to be an outlier. Further investigations are needed to study the cause of this outlier. As mentioned in Analysis of Variance Approach , ANOVA, a perfect regression model results in a fitted line that passes exactly through all observed data points. Thus, no error exists for the perfect model. The deviations in observations recorded for the second time constitute the "purely" random variation or noise. The data is collected as shown next:.

One portion is the pure error due to the repeated observations. The other portion is the error that represents variation not captured because of the imperfect model. Thus, for an imperfect regression model:. The test statistic for the lack-of-fit test is:.

Assume that a second set of observations are taken for the yield data of the preceding table. The resulting observations are recorded in the following table.

Therefore, at a significance level of 0. The linear regression model may not be directly applicable to certain data. Non-linearity may be detected from scatter plots or may be known through the underlying theory of the product or process or from past experience.