Mathematics
Hypothesis Test for Regression Slope
A hypothesis test for regression slope is a statistical method used to determine if there is a significant linear relationship between two variables in a regression model. It involves testing the null hypothesis that the slope of the regression line is equal to zero, indicating no relationship, against the alternative hypothesis that the slope is not equal to zero, suggesting a significant relationship.
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12 Key excerpts on "Hypothesis Test for Regression Slope"
- Nigel Walford(Author)
- 2011(Publication Date)
- Wiley(Publisher)
(X). The fate of the Null Hypothesis is judged by specifying a level of significance (e.g. 0.05) and determining the probability of having obtained the value of the F statistic. If the probability is less than or equal to the chosen significance threshold, the Null Hypothesis would be rejected.The alternative method of testing the significance of regression analysis is to focus on the individual elements in the regression equation, the slope and intercept constants and the predicted Y values, which estimate the mean of the dependent variable for each known value of X . The procedure is similar in each case and recognizes that if an infinite number of samples were to be selected from a population there would be a certain amount of dispersion in the slope, intercept and predicted Y values (for each value of X). This can be quantified by means of calculating the standard error, which is used to produce t test statistics and confidence limits in each case. The general form of the Null Hypothesis for each of these tests is that any difference between the sample-derived statistics (b , a and ŷ ) and the known or more likely assumed population parameters (β α and µ ) has arisen through sampling error. There are n – 2 degrees of freedom when determining the probability associated with the t test statistic calculated from the sample data values. As usual, the decision on whether to accept the Null Hypothesis or the Alternative Hypothesis is based on the probability of the t test statistic in relation to the stated level of significance (e.g. 0.05).The calculation of confidence limits for the slope, intercept and predicted Y values provide a range of values within which the corresponding population parameter is likely to lie with a certain degree of confidence (e.g. 95% or 99%). In each case the standard error is multiplied by the t statistic value for the chosen level of confidence with n – 2 degrees of freedom and the result is added to and subtracted from the corresponding sample derived statistic in order to define upper and lower limits. In the case of confidence limits for the predicted Y values, these can be calculated for several known values of X
eBook - ePub- (Author)
- 2020(Publication Date)
- Wiley(Publisher)
To understand the concepts involved in this test, it is useful to first review a simple, equivalent approach based on confidence intervals. We can perform a hypothesis test using the confidence interval approach if we know three things: (1) the estimated parameter value, or, (2) the hypothesized value of the parameter, b 0 or b 1, and (3) a confidence interval around the estimated parameter. A confidence interval is an interval of values that we believe includes the true parameter value, b 1, with a given degree of confidence. To compute a confidence interval, we must select the significance level for the test and know the standard error of the estimated coefficient. Suppose we regress a stock’s returns on a stock market index’s returns and find that the slope coefficient () is 1.5, with a standard error of 0.200. Assume we used 62 monthly observations in our regression analysis. The hypothesized value of the parameter (b 1) is 1.0, the market average slope coefficient. The estimated and population slope coefficients are often called beta, because the population coefficient is often represented by the lowercase Greek letter beta (β) rather than the b 1 that we use in our coverage. Our null hypothesis is that b 1 = 1.0 and is the estimate for b 1. We will use a 95 percent confidence interval for our test, or we could say that the test has a significance level of 0.05. Our confidence interval will span the range to, or where t c is the critical t -value (note that we use the t -distribution for this test because we are using a sample estimate of the standard error, s b, rather than its true population value). The critical value for the test depends on the number of degrees of freedom for the t -distribution under the null hypothesis. The number of degrees of freedom equals the number of observations minus the number of parameters estimated
eBook - ePub- Alan Anderson(Author)
- 2023(Publication Date)
- For Dummies(Publisher)
b is the slope. β ≠ 0 and ρ ≠ 0 is a reminder that this is a two-tailed test.t is the test statistic for testing the hypothesis that the slope coefficient of the regression line equals zero. p is the p-value for this test; because it is less than the level of significance of 0.05, the null hypothesis can be rejected. This means that X does explain Y ; in other words, studying time does help explain a student’s GPA. df refers to the degrees of freedom for this test; this equals n − 2 (in this case, 8 − 2 = 6).a = 1.05 and b = 0.15 are the intercept and slope coefficient, respectively, of the estimated sample regression line. The resulting sample regression line can be expressed as: Y = 1.05 + 0.15X.s is the standard error of the estimate (SEE) for this regression equation. It can be thought of as the common standard deviation shared by all residuals (estimated errors) for this estimated regression equation.is the R-squared measure; because it is close to 1, this indicates a good fit of the regression model to the sample data.r is the correlation coefficient between X and Y . (Note that this equals the square root of .) In this case, the correlation is about 0.89. Because the correlation coefficient cannot exceed 1 (or fall below −1), a correlation of 0.89 indicates a very strong, positive relationship between X and Y . (Correlation is discussed in Chapter 5 .) In other words, there is a strong relationship between the amount of time a student spends studying each month and the student’s GPA.Note that it is possible to compute the sample regression equation without any of the corresponding statistical measures. After entering the X (L2) and Y (L1) data into lists, follow these steps:
eBook - PDF- Myles Hollander, Douglas A. Wolfe, Eric Chicken(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
Chapter 9 Regression Problems INTRODUCTION Among the most common applications of statistical techniques are those involving some sort of regression analysis. Such procedures are designed to detect and interpret stochastic relationships between a dependent (response) variable and one or more independent (predictor) variables. These regression relationships can vary from that of a simple linear relationship between the dependent variable and a single independent variable to complex, nonlinear relationships involving a large number of predictor variables. In Sections 9.1–9.4, we present nonparametric procedures designed for the simplest of regression relationships, namely, that of a single stochastic linear relationship between a dependent variable and one independent variable. (Such a relationship is commonly referred to as a regression line .) In Section 9.1, we present a distribution-free test of the hypothesis that the slope of the regression line is a specified value. Sections 9.2 and 9.3 provide, respectively, a point estimator and distribution-free confidence intervals and bounds for the slope parameter. In Section 9.4, we complete the analysis for a single regression line by discussing both an estimator of the intercept of the line and the use of the estimated linear relationship to provide predictions of dependent variable responses to additional values of the predictor variable. In Section 9.5, we consider the case of two or more regression lines and describe an asymptotically distribution-free test of the hypothesis that the regression lines have the same slope; that is, that the regression lines are parallel. In Section 9.6, we present the reader with an introduction to the extensive field of rank-based regression analysis for more complicated regression relationships than that of a straight line.
- Luther Tweeten(Author)
- 2019(Publication Date)
- Routledge(Publisher)
Alternatively stated, the test is to determine if the estimated coefficient is significantly different from zero. If the statistical testing procedure suggests rejection of this null hypothesis, the coefficient is said to be statistically different from zero (at a specified significance level). If the procedure gives the contrary result, we fail to reject the null hypothesis. (Technical reasons involving statistical theory and rules of logic preclude using the term accept in place of fail to reject. See Henkel, pp. 40-41; Tweeten, p. 548.) The alternative hypothesis could be nondirectional as in equation 2.29 or it may be directional as in equation 2.32. With a directional alternative hypothesis, a rejection of the null hypothesis is in favor of the alternative hypothesis in the indicated direction. For example, if the alternative hypothesis were as follows: Alt. Hyp: Bk < 0 (2.32) then rejecting the null hypothesis suggests that ~k is not only significantly different from zero, it also has a negative sign. This test situation, or with the inequality reversed in the alternative hypothesis, is referred to as a one-tailed test The alternative hypothesis chosen for the test is usually determined by the amount of information the researcher has about the relationship being modeled. For example, if a supply equation is being estimated, the researcher will, based on theory and review of literature, be in a position to specify the signs of the coefficients. The parameter attached to the commodity's own price is expected to be positive; coefficients on prices of other commodities that compete for production resources are expected to be negative. In this case, with a null hypothesis equal to zero, the alternative hypothesis is for the coefficient on own price to be greater than zero and for each of the alternative hypotheses relating to the other prices to be negative.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
Finance The capital asset pricing model uses linear regression as well as the concept of Beta for analyzing and quantifying the systematic risk of an investment. This comes directly from the Beta coefficient of the linear regression model that relates the return on the investment to the return on all risky assets. Regression may not be the appropriate way to estimate beta in finance given that it is supposed to provide the volatility of an investment relative to the volatility of the market as a whole. This would require that both these variables be treated in the same way when estimating the slope. Whereas regression treats all variability as being in the investment returns variable, i.e. it only considers residuals in the dependent variable. Environmental science Linear regression finds application in a wide range of environmental science applications. In Canada, the Environmental Effects Monitoring Program uses statistical analyses on fish and benthic surveys to measure the effects of pulp mill or metal mine effluent on the aquatic ecosystem. Statistical hypothesis testing A statistical hypothesis test is a method of making decisions using experimental data. In statistics, a result is called statistically significant if it is unlikely to have occurred by chance. The phrase test of significance was coined by Ronald Fisher: Critical tests of ________________________ WORLD TECHNOLOGIES ________________________ this kind may be called tests of significance, and when such tests are available we may discover whether a second sample is or is not significantly different from the first. Hypothesis testing is sometimes called confirmatory data analysis , in contrast to exploratory data analysis.
- Samprit Chatterjee, Jeffrey S. Simonoff(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
In this situation, it is likely that the ‐statistic for each predictor will be relatively small. This is not an inappropriate result, since given one predictor the other adds little (being highly correlated with each other, one is redundant in the presence of the other). This means that the ‐statistics are not effective in identifying important predictors when the two variables are highly correlated. The ‐tests and ‐test of Section 1.3.3 are special cases of a general formulation that is useful for comparing certain classes of models. It might be the case that a simpler version of a candidate model (a subset model) might be adequate to fit the data. For example, consider taking a sample of college students and determining their college grade point average (), Scholastic Aptitude Test (SAT) evidence‐based reading and writing score (), and SAT math score (). The full regression model to fit to these data is Instead of considering reading and math scores separately, we could consider whether can be predicted by one variable: total SAT score, which is the sum of and. This subset model is with. This equality condition is called a linear restriction, because it defines a linear condition on the parameters of the regression model (that is, it only involves additions, subtractions, and equalities of coefficients and constants). The question about whether the total SAT score is sufficient to predict grade point average can be stated using a hypothesis test about this linear restriction. As always, the null hypothesis gets the benefit of the doubt; in this case, that is the simpler restricted (subset) model that the sum of and is adequate, since it says that only one predictor is needed, rather than two. The alternative hypothesis is the unrestricted full model (with no conditions on). That is, versus These hypotheses are tested using a partial ‐test
eBook - ePub- Samprit Chatterjee, Jeffrey S. Simonoff(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
model selection.First, we discuss the uses of hypothesis testing for model selection. Various hypothesis tests address relevant model selection questions, but there are also reasons why they are not sufficient for these purposes. Part of these difficulties is the effect of correlations among the predictors, and the situation of high correlation among the predictors (collinearity) is a particularly challenging one.A useful way of thinking about the tradeoffs of overfitting versus underfitting noted above is as a contrast between strength of fit and simplicity. The principle of parsimony states that a model should be as simple as possible while still accounting for the important relationships in the data. Thus, a sensible way of comparing models is using measures that explicitly reflect this tradeoff; such measures are discussed in Section 2.3.1.The chapter concludes with a discussion of techniques designed to address the existence of well-defined subgroups in the data. In this situation it is often the case that the effects of a predictor on the target variable is different in the two groups, and ways of building models to handle this are discussed in Section 2.4.2.2 Concepts and Background Material
2.2.1 USING HYPOTHESIS TESTS TO COMPARE MODELS
Determining whether individual regression coefficients are statistically significant as discussed in Section 1.3.3 is an obvious first step in deciding whether a model is overspecified. A predictor that does not add significantly to model fit should have an estimated slope coefficient that is not significantly different from 0, and is thus identified by a small t-statistic. So, for example, in the analysis of home prices in Section 1.4, the regression output on page 16 suggests removing number of bedrooms, lot size, and property taxes from the model, as all three have insignificant t
- David Howell(Author)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
252 Chapter 10 Regression By clicking on the point at approximately (2.5, 26.5) I have changed the slope of the regression line from –3.0 to –3.3. The less steep line in this figure is the line for all 24 observations, whereas the steeper line fits the data, omitting the point on which I clicked. You can click on each data point and see the effect on the slope and intercept. One of the points in this chapter concerned the use of Student’s t test to test the null hypothesis that the true slope in the population is 0.00 (i.e., the hypothesis that there is no linear relationship between X and Y .) The applet named SlopeTest illustrates the meaning of this test with population data that support the null hypoth-esis. A sample screen is illustrated here. In this applet I have drawn 100 samples of five pairs of scores. I drew from a population where the true slope, and therefore the true correlation, is 0.00, so I know that the variables are not linearly related. For each of the 100 slopes that I obtained, the applet calculated a t test using the formula for t given in Section 10.6. The last 10 t values are given near the top of the display, and they range from –2.098 to 5.683. The distribution of all 100 values is given at the right, and the plot of the five observations for my 100th set is given at the left. (The green line on the left always has a slope of 0 and represents what the line would be with a huge sample with the null hypothesis being true.) Each time I click the “10 sets” button, I will draw 10 new sets of observa-tions, calculate their slopes and associated t values, and add those to the plot on the right. If I click the “100 Sets” button, I will accumulate 100 t values at a time. Run this applet. First generate one set at a time and note the resulting variation in t and how the regression line changes with every sample. Then accumulate 100 sets at a time and notice how the distribution of t smoothes out.
eBook - ePub- K. Nirmal Ravi Kumar(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
This r 2 is significant, as indicated by ‘F cal ’ value, that is significant at 5 per cent level (‘P’ value (0.02) < α (0.05)). So, it can be concluded that, ‘ we reject the H O and conclude that, there was a positive and significant relationship between fertilizer application and output of a crop. Furthermore, 80 per cent of the variability in output could be explained by fertilizer application ’. III. Tests of Significance in Regression : Note that, there are several hypotheses that are to be tested in regression analysis: • That the variation explained by the regression model is not due to chance ie., to study the significance of overall regression model (F test). • That the slope of the regression line (b ^) is significantly different than zero ie., to study the significance of individual regression coefficient estimate (‘t’ test). • That the Y intercept (ie., â) is significantly different than zero (‘t’ test). These tests are discussed in-detail in the ensuing Chapters 4 and 9 of SLRM and MLRM respectively. In the above regression model, if we consider another independent variable say, irrigation cost (X 2), it represents MLRM. It is essential to mention that, the main tool of econometrics is the linear (multiple) regression model. This is because, a regression seeks to estimate the marginal impact of a particular independent variable after taking into account the impact of the other independent variables in the model. For example, the MLRM model may try to isolate the effect of a one percentage point increase in fertilizer application on average output produced, holding constant other determinants of output like, pesticides cost, irrigation cost, human labour cost etc
eBook - PDF- Ned Freed, Stacey Jones, Timothy Bergquist(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
CHAPTER Basic Regression Analysis LEARNING OBJECTIVES After completing the chapter, you should be able to 1. Describe the nature and purpose of regression analysis. 2. Calculate the slope and the intercept terms for a least squares line. 3. Compute and interpret measures of fit, including the standard error of estimate, coefficient of determination, and correlation coefficient. 4. Discuss the inference side of regression and summarize the basic assumptions involved. 5. Build interval estimates of the slope and intercept terms in a regression equation. 6. Conduct a proper hypothesis test for the slope term in a regression equation and interpret the result. 7. Estimate expected and individual values of y in regression. 8. Read and interpret a computer printout of results from a simple linear regression analysis. 9. Check errors (or residuals) to identify potential violations of the assumptions in regression. 11 © Alex Millos/Shutterstock 11.1 An Introduction to Regression 389 W e’ve all seen them. Those ’breakthrough’ headlines that start with “Researchers find link between…” or “New study connects ’X’ to …” In the past year alone, we’ve been told that drinking coffee leads to a longer life, prevents Alzheimer’s disease, lowers the risk of skin cancer, lessens the probability of heart failure, causes heart failure, decreases the chance of stroke, and leads to vision loss. We’ve been assured that eating pizza reduces the risk of colon cancer, anchovies promote weight loss, and studying makes you nearsighted. But before you call Domino’s to order that pizza with extra anchovies, you need to be aware of one important fact: The research behind most of these bold “new study” head- lines establishes correlation, not causation. Correlation is simply an indicator of how two sets of values appear to vary together. Eating pizza might be correlated with a lower incidence of cancer, but that doesn’t mean you should eat pizza for breakfast, lunch and dinner.
eBook - ePub- Danny McCarroll(Author)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
It is often useful to compare two regression analyses to see if they are significantly different. Common examples in geography dissertations involve comparing relationships in one place with those in another place, or comparing relationships measured in a system (e.g. a river) that has been disturbed in some way with the same relationships measured in a ‘control’ system. I have seen many dissertations that try to do this but they rarely use the data that they have collected to best effect. This kind of comparison is quite complicated and you need to be very clear about the differences you are testing for. You can test whether there is a significant difference between the two correlation coefficients, between the two slope values, or for a difference in the two sets of measured values. An example is perhaps the easiest way to explain.Imagine that you have measured the average size of the stones forming the bedload at 15 points down a river. The local bedrock comprises mudstone and limestone and you measure the size of the two rock types separately (Figure 11.16 ). You wish to know whether the change in size with distance is the same for the two rock types.The first question you might ask is whether the relationship between distance and size is equally strong for both rock types. The appropriate test here is for the significance of the difference between two correlation coefficients (Section 10.6 ). In this case the two correlation coefficients are –0.966 and –0.925 (the square root of the two R 2 values with the sign added) and they are not significantly different (z = –0.75, two-tail p > 0.05). We can conclude that the relationship between size and distance is equally strong for the two rock types.Both rock types show a decline in size with distance, but note that the rate of change is not the same. The slope coefficient of the equation describes the decline in size per unit distance and is 81 mm per km for mudstone whereas for limestone it is only 17.6 mm per km. The two slope coefficients can be compared using a t -test. If we call the two regression equations A and B then the parameters we need areFIGURE 11.16
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