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.
7 Key excerpts on "Hypothesis Test for Regression Slope"
eBook - ePubStatistics for the Behavioural Sciences
An Introduction to Frequentist and Bayesian Approaches
- Riccardo Russo(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...In this case we would have s 2 x = s x = s y = 1 and r = b. It therefore appears that the correlation coefficient measured on standardised data and the standardised regression coefficient (i.e., the slope obtained on standardised data) are identical. 11.4 Hypothesis testing on the slope b In estimating the slope b we usually work with samples. It is possible that, by chance, we obtain a value of b larger (or smaller) than zero, even if the slope obtained by regressing variable Y on X in the population, i.e., β, is zero (if β = 0 it means that in the population there are no linear increments (or decrements) in the dependent variable as a function of increments in the independent variable). Hence, like the correlation coefficient r, it is useful to test whether the value of b is significantly different from zero. Consider also that if it is true that X and Y are linearly correlated, then it should also be true that the slope of the regression line obtained by regressing Y on X is different from zero. Moreover, we saw that r and b are strictly related since both their formulae have the covariance between X and Y as their numerator. It then follows that if this covariance is zero, r = b = 0. Hence, testing for the significance of b is equivalent to testing for the significance of r. In testing the significance of b we assume that the following null hypothesis is true: H 0 : β = 0 (where β indicates the slope in the population). The alternative hypothesis is H 1 : β ≠ 0. To test this null hypothesis we need to know the standard error of b. Assuming that in the population, for each value of the variable X, the marginal distributions of the values of the variable Y, around the regression line, are normal with homogeneous variances (see Figure 11.3). Then it can be shown that the quantity b s y s x × 1 − r 2 n − 2 (its denominator is the standard error of b : S E b = s y s x × 1 − r 2 n − 2) is distributed as t with n − 2 degrees of freedom...
eBook - ePub- James P. Neelankavil(Author)
- 2015(Publication Date)
- Routledge(Publisher)
...When the sample parameter b is known, the model can be evaluated by estimating the confidence interval of the slope b. To evaluate a regression equation, a test of its slope will provide the necessary results. That is, if the data points fall close to a linear pattern, the slope is assumed to be nonzero and therefore a good linear fit. If the data points are widely dispersed about the regression line, the fit is poor and the slope near 0 (Figure 16.4). Figure 16.3 Relationship Between X and Y For each value of X, the distribution of the points about the regression (mean of a + bX) is normal with mean 0 and a constant variance of σ 2 (as shown in Figure 16.4). The approximate regression equation for the sample is represented by Y i = a 1 + b 1 X (the regression equation for the total population would be Y = a + bX) To determine whether the slope is significantly different from 0, a t test could be applied. The following formulas are applied to test the regression equation (the derivation of the formula is complicated and not necessary for understanding the use of the formula). where S 2 Y.X = the variance of regression (the standard deviation is the square root of this). This can be computed by The degree of freedom is n − 2 because the two parameters b (actual slope) and b 1 (assumed slope) are approximated in order to compute S Y.X. Figure 16.4 As discussed in chapter 14, to estimate the confidence interval, a test of hypotheses is carried out. Using example 2 (trust in the cosmetics company), we can demonstrate the confidence interval for the regression using hypothesis testing. Let us assume a 2.5 percent level of significance that the trust in the company is linearly related to the amount spent on its brand of cosmetics...
eBook - ePub- Gary Rassel, Suzanne Leland, Zachary Mohr, Elizabethann O'Sullivan(Authors)
- 2020(Publication Date)
- Routledge(Publisher)
...And you might rightfully wonder whether the finding of significance or lack of significance was associated with how the values were combined. Alternatively, you could hypothesize that the average achievement test grades at School A are higher than at School B. You could test that hypothesis using a t -test. The t -test is an interval statistic that can test hypotheses that two groups have different means. Each group is considered a sample, and the appropriate test is a two-sample t -test. The two-sample t -test can test a research hypothesis (1) that two groups have different means or (2) that one group’s mean is higher than the other’s. The first type of hypothesis specifies no direction; a two-tailed t -test is used to test nondirectional hypotheses. The second type of hypothesis has direction; a directional hypothesis is tested with a one-tailed t -test. The degree of variability in each sample affects the choice of formula used to calculate the t -value. One formula is used if the variances of each population represented by the samples are assumed to be equal. Another formula is used if the populations are assumed to have unequal variances. A third formula is used if data for both groups come from the same subjects. For example, a researcher may compare analysts’ error rates when entering data on laptop computers as opposed to desktop computers. A single group of analysts would enter data into both laptop and desktop computers. The dependent variable would be error rate, and the independent variable would be type of computer. Other examples include measures on the same individuals taken before and after some event. The t -test can also test a single sample hypothesis, for example, that a group’s mean is greater or less than a specified value. The single sample t -test is useful for various situations such as staffing studies. 9 For example, a university library may assume that its reference desk handles an average of 25 inquiries per hour on weekends...
eBook - ePubSocial Statistics
Managing Data, Conducting Analyses, Presenting Results
- Thomas J. Linneman(Author)
- 2017(Publication Date)
- Routledge(Publisher)
...Chapter 8 Using Sample Slopes to Talk About Populations Inference and Regression This chapter covers.... . . one last probability distribution, this time using sample slopes. . . what the standard error of the slope really means. . . how to test a slope for statistical significance. . . the role of sample size in testing a slope for significance. . . how a researcher used regression to study the relationship between housing appreciation and support for social security. . . how a researcher used regression to study how family size and grades are related Introduction The goal of this chapter is to take the inferential techniques we’re already covered and apply them to the regression slopes you learned about in the previous chapter. Given that you are a seasoned pro at inference by this point, you will undoubtedly experience some déjà vu (or because you’re reading this, déjà lu). Inferential techniques have a lot in common with one another, so once you learn the overall goal of inference, it’s really just a variation on a theme. The goal of inference with slopes is to be able to claim that, in the population, the independent variable has an effect on the dependent variable. One More Sampling Distribution Chapter 4 included a distribution of hundreds of chi-square values. Chapter 5 had a distribution of hundreds of sample means. Chapter 6 involved a sampling distribution of sample mean differences. Here, I do roughly the same thing, but instead of chi-square values, or sample means, or differences between sample means, my building blocks will be sample slopes. I went back to my original hypothetical dataset of 100 grades I used in previous chapters. To each student’s grade, I added another piece of information: the percentage of classes that student attended during the semester...
eBook - ePubSocial Statistics
Managing Data, Conducting Analyses, Presenting Results
- Thomas J. Linneman(Author)
- 2021(Publication Date)
- Routledge(Publisher)
...Chapter 8 Using Sample Slopes to Talk About Populations Inference and Regression DOI: 10.4324/9781003220770-8 This chapter covers … … one last probability distribution, this time using sample slopes … what the standard error of the slope really means … how to test a slope for statistical significance … the role of sample size in testing a slope for significance … how a researcher used regression to study the relationship between housing appreciation and support for social security … how researchers used regression to study skin color and educational attainment Introduction The goal of this chapter is to take the inferential techniques we’re already covered and apply them to the regression slopes you learned about in the previous chapter. Given that you are a seasoned pro at inference by this point, you will undoubtedly experience some déjà vu (or because you’re reading this, déjà lu). Inferential techniques have a lot in common with one another, so once you learn the overall goal of inference, it’s really just a variation on a theme. The goal of inference with slopes is to be able to claim that, in the population, the independent variable has an effect on the dependent variable. One More Sampling Distribution Chapter 4 included a distribution of hundreds of chi-square values. Chapter 5 had a distribution of hundreds of sample means. Chapter 6 involved a sampling distribution of sample mean differences. Here, I do roughly the same thing, but instead of chi-square values, or sample means, or differences between sample means, my building blocks will be sample slopes. I went back to my original hypothetical dataset of 100 grades I used in previous chapters. To each student’s grade, I added another piece of information: the percentage of classes that student attended during the semester...
eBook - ePub- K. Nirmal Ravi Kumar(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
...We know, ‘r’ is a sample correlation coefficient and it serves as an estimate of population correlation coefficient ‘ρ’. As a statistical estimate, ‘r’ is subjected to error because of sampling and hence, it should be tested for validity. Student’s ‘t’ test will be employed to test the significance of ‘r’ value. This test provides the researcher with some evidence that, there really is a relationship between the two variables and not due to peculiarity in the data. In order to test the correlation coefficient for statistical significance, it is necessary to define the true correlation coefficient that would be observed, if all population values were obtained. As mentioned earlier, this true correlation coefficient is usually given the Greek symbol ‘r’ (rho). So, the null hypothesis (H O) is that: H O : ρ = 0 (there is no relationship between the two variables X and Y, when all population values are observed) and an alternative hypothesis (H A) is framed against this as: H A : Alternative hypothesis could be any one of the following three forms viz., (ρ≠0) or (ρ<0) or (ρ>0). That is, if the researcher has no idea whether or how two variables are related, then a two tailed H A ie., H A : (ρ≠0) is formulated. If the researcher suspects, or has knowledge, that the two variables are negatively related then, H A : (ρ<0) is formulated (ie., one tailed H A) and if the researcher predicts a positive relationship between the variables, then H A : (ρ>0) is formulated (ie., one tailed H A). The test statistic for the hypothesis test is the sample or observed correlation coefficient ‘r’ obtained from a sample. The sampling distribution of ‘ r ’ is approximated by a ‘ t ’ distribution with n –2 Degrees of Freedom (df). The formula to compute ‘ t cal ’ value is given by: t = r S E (r) Equation 2.14 where, r = Sample correlation coefficient, Standard Error (SE) of ‘ r ’ = 1 − r 2 n − 2 n = number of observations...
eBook - ePubStatistics in Social Work
An Introduction to Practical Applications
- Amy Batchelor(Author)
- 2019(Publication Date)
- Columbia University Press(Publisher)
...When using regression analysis, you are limited in the same ways as you are for any type of inferential statistics. You have selected data relevant to the question you are trying to answer, but those data only represent a certain group of people at a certain time. You still have to determine if the results can be applied to other populations. When you come across a regression analysis in the literature, you should check for a few things before deciding whether to use this information in your practice. 1. Is the relationship linear? If the pattern created by the data points does not generally follow a straight line, then linear regression is not appropriate. 2. Are all important variables controlled for? If age is likely to have a big effect on any health outcomes of interest, does the study control for age? What about income? Are there other factors that could contribute to this relationship that have not been addressed? 3. Is the sample used in the analysis similar to the people you work with? The results of these analyses are only applicable to samples that are similar. If you work with children, a sample made up of adults is unlikely to be useful to you. If you work with people who live in cities, a sample of rural farmers is unlikely to be useful to you. 4. Remember that correlation is not causation. Regression analysis cannot tell you that one variable caused an effect in the other. You may find a statistically significant relationship between two variables that have nothing to do with each other but happen to behave in similar ways. Beware of spurious correlations. MORE ON HYPOTHESES: ONE- AND TWO-TAILED TESTS Returning to the concept of hypothesis testing from chapter 7, you can further refine your practice by understanding the differences between one- and two-tailed tests. For either a one- or two-tailed test, assume you are still using the significance level of.05...
