Paired T-Test

11 Key excerpts on "Paired T-Test"

• 

  eBook - ePub

  Medical Statistics

  A Guide to SPSS, Data Analysis and Critical Appraisal

  • Belinda Barton, Jennifer Peat(Authors)
  • 2014(Publication Date)
  • BMJ Books

    (Publisher)

  Chapter 4 Paired and one-sample t-tests

  A statistician is a person who likes to prove you wrong, 5% of the time.

  TAKEN FROM AN INTERNET BULLETIN BOARD

  Objectives

  The objectives of this chapter are to explain how to:

  • analyse paired or matched data
  • use paired t-tests and one-sample t-tests
  • interpret results from non-parametric paired tests
  • calculate an effect size
  • report changes or differences in paired data in appropriate units

  In addition to two-sample (independent) t-tests, there are also two other parametric t-tests that can be used to analyse continuous data, that is, paired t-tests and one-sample (single sample) t-tests. All three types of t-test can be one-tailed or two-tailed tests. However, one-tailed t-tests are rarely used in health sciences research.

  4.1 Paired t-tests

  A paired t-test is used to estimate whether the means of two related measurements are significantly different from one another. This test is used when two continuous variables are related because they are collected from the same participant at different times, from different sites on the same person at the same time or from cases and their matched controls.1 Examples of paired study designs are

  • data from a longitudinal study;
  • measurements collected before and after an intervention in an experimental study;
  • differences between related sites in the same person, for example limbs, eyes or kidneys;
  • matched cases and controls.

  For a paired t-test, there is no explanatory (group) variable. The outcome of interest is the difference in the outcome measurements between each pair or between each case and its matched control, that is, the within-pair differences. When using a paired t-test, the variation between the pairs of measurements is the most important statistic and the variation between the participants, as when using a two-sample t-test, is of little interest. The null hypothesis for a paired t

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Introductory Statistics for the Health Sciences

  • Lise DeShea, Larry E. Toothaker(Authors)
  • 2015(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  313 Formula for the Paired t Test In sum, both formulas follow the pattern of “(something minus its mean) divided by its estimated standard deviation.” For the paired t test, the “some-thing” is the mean of the difference scores (or, equivalently, the difference in the means for the two conditions). “Its mean” comes from the null hypothesis, where the μ d was zero. “Its estimated standard deviation” is the estimated standard error of the paired mean difference (i.e., the estimated standard error of a sam-pling distribution for the difference in the two paired means). The last sentence is a good reminder that every statistic has a sampling distri-bution, so the paired t test does too. That sentence also needs some explanation: we could imagine drawing all possible samples of the same size from the same population, repeating the study of the pain ratings for shots with and without vibration, computing the paired t test on each sample’s data, and arranging those paired t tests’ numeric results in a distribution. What would the sampling dis-tribution look like? If the null hypothesis were true and the vibration made no difference, and if two assumptions are met, then the distribution would look like a theoretical t distribution. We just mentioned that the paired t test has two assumptions. The assump-tions are the following: • Pairs of scores (or the d ’s) are normally distributed in the population. • Pairs of scores (or the d ’s) are independent of each other. Clearly there is a connection within each pair of scores: either the same per-son is being measured twice, or we are measuring two people who have either a natural connection (like twins) or a researcher-created connection. In the case of the twins being compared on bone mineral density, one twin is not indepen-dent of the other—but that is not what the assumption of independence says. The assumption is that each pair is independent of every other pair.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Statistics for the Social Sciences

  A General Linear Model Approach

  • Russell T. Warne(Author)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  9 Paired Two-Sample t-Tests By now we have mastered two types of null hypothesis statistical significance tests (NHSTs) – the one-sample t-test and the z-test. Both have the same null hypothesis, which is that the sample mean " Y ð Þ is equal to the population mean (μ y ). Mathematically, we expressed this null hypothesis as H 0 : " Y ¼ μ y . Both of these NHST procedures are useful for determining whether the sample mean is statistically equal to the population mean. However, that is rarely the research question that social scientists, practitioners, teachers, and therapists ask. Often the reason is that they do not know the population mean. Instead, they are frequently interested in whether two groups of sample members are similar or different. In Chapters 9 and 10 we will discuss two NHST procedures that answer this question. The first procedure – which this chapter covers – is called the paired two-sample t-test. This is useful for (1) investigating growth and change, and (2) comparing samples of data that are linked in some way. Learning Goals After studying this chapter you should be able to: • Find appropriate uses for a paired two-sample t-test NHST procedure. • Conduct a paired two-sample t-test. • Calculate an effect size for a paired two-sample t-test. When to Use the Paired Two-Sample t-Test There are three situations in which it is appropriate to use the paired two-sample t-test. The first occurs when the investigator is interested in measuring growth or change. An example of this is a study conducted by Diefenbach et al. (2007). These researchers wondered whether patients with obsessive-compulsive disorder had a better quality of life after psychological therapy than they did before therapy. This required giving their patients a quality of life survey before they started therapy and then administering the same quality of life survey after therapy was finished.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Statistics for the Social Sciences

  A General Linear Model Approach

  • Russell T. Warne(Author)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  Second, independent researchers have developed the same statistical procedures inde- pendently and without being aware of others’ work (e.g., see Conroy, 2012). As a result, statistical terminology that one author uses depends on which “discoverer” the author used as their source for terminology. The paired-samples t-test is an example of how terminology can be confusing. Two common alternative names for the paired samples t-test are the “dependent samples t-test” and the “two-sample dependent t-test.” I avoid this terminology because I think that the term “dependent” is easily confused with the term “dependent variable,” which is the outcome variable (see Chapter 1). Another term is the “related-samples t-test,” which I avoid in this book because it can be confused with the term “relation- ship,” which describes how scores on different variables appear together (see Chapters 7 and 12). In this textbook I have tried to use the terminology that is most widespread in psychology, sociology, family science, anthropology, social work, and education. Most of this terminology is also applicable to other related fields, such as communications and business. (On the other hand, researchers in economics sometimes use a different terminology from that in the other social sciences.) Because you may encounter other terms for the same procedures or ideas, I have put some alternative terms in the glossary. These will direct you to the equivalent terminology used in this textbook. 216 Paired-Samples t-Tests 1. Form groups in the data. 2. Define the null hypothesis (H 0 ). The null hypothesis is always that there is no difference between groups or that there is no relationship between independent and dependent variables. 3. Set alpha (α). The default alpha = .05. 4. Choose a one-tailed or a two-tailed test. This determines the alternative hypothesis (H 1 ). 5. Find the critical value, which is used to define the rejection region.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Statistics for Anthropology

  • Lorena Madrigal(Author)
  • 2012(Publication Date)
  • Cambridge University Press

    (Publisher)

  However, the t test is so popular that it deserves to be in its own chapter. This test is also frequently known as “Student’s t test” because the t distribution was first proposed by Gosset in a paper which he authored under the pseudonym “Student” (Fienberg and Lazar 2001). The t test is used if a researcher needs to compare two samples to determine if they differ significantly. Examples in anthropology are the mean age at marriage of two ethnic groups, the mean length of projectile points, the mean height of two groups of children, etc. The null and alternative hypotheses in a two-tailed test are H0:  1 =  2 , and H1:  1 =  2 . A one-tailed test should only be done (as always) when the researcher knows, 5.1 The un-paired t test 109 or is interested in testing only that one group has a greater or smaller mean than the other. In a one-tailed test, the null and alternative hypothesis are H0:  1 ≥  2 , and H1:  1 <  2 , or H0:  1 ≤  2 , and H1:  1 >  2 , depending on the direction of the treatment effect. The problem when comparing two sample means is that we need to agree on how different two means have to be for us to decide that they were not sampled from the same population. As we learned in chapter four, means of sample size n will be approximately normally distributed around the parametric mean. Thus, some means from the same population may be different enough from each other for us to decide (in error) that they were not sampled from the same population. On the other hand, some means do not differ at all and their difference is 0, which is the mean of the distribution of sample-mean differences. The more different the two samples are, the larger the difference between the means will be. As the difference increases in value it will be farther apart from the mean of sample differences ( = 0), and it will have a lower frequency because it will be located towards the tails of the distribution.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Statistics for the Behavioural Sciences

  An Introduction

  • Riccardo Russo(Author)
  • 2004(Publication Date)
  • Psychology Press

    (Publisher)

  Chapter 9 Comparing a pair of means

  The matched- and the independent-samples t-test

  Introduction

  In Chapter 8 we mainly described how to decide if a set of observations was drawn from a given population. We showed how to apply the t-test in assessing the probability that a set of observations, with their observed mean, was drawn from a population with a given mean. In that chapter we also showed how to use sample data to construct confidence intervals that have a given probability of containing the true population mean. In this chapter we will extend the application of the t-test to situations in which we intend to compare the means obtained from two sets of observations. The aim of this comparison is to decide if the two sets are sampled from the same population or not.

  If the outcome of the t-test gives a sufficiently small probability of the two sets of observations being drawn from the same population, then we declare that the two sets of observations are drawn from different populations (i.e., there is a significant difference between the means). Two different conditions in which the t-test is applied to compare a pair of means will be described:

  1. when the means are obtained from two sets of related observations; and
  2. when the two sets of observations are independent.

  We also explain how to construct confidence intervals for the difference of a pair of population means, and we describe the conditions where the t-test may not be appropriate for comparing pairs of means. This occurs when some of the assumptions underlying the use of the t-test are violated. For these cases we indicate alternative ways to analyse the data. Given the strong similarity to the one-sample t-test, we first describe the application of the t

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Statistical Reasoning in Law and Public Policy

  Tort Law, Evidence and Health

  • Joseph L. Gastwirth(Author)
  • 1988(Publication Date)
  • Academic Press

    (Publisher)

  590 Statistical Reasoning in Law and Public Policy respect to many factors, as the size of each strata would become quite small. If that happens, almost all of the observations in some strata may come from only one of the two populations being compared, thereby decreasing the precision of the estimated difference and the power of the test to determine its statistical significance. 2. Statistical Inference From Paired Data Using the Average Difference Between the Matched Pairs: The t-Test The data available to us in a matched pair analysis consist of pairs (JC,, y,) i = 1, ..., n of measurements, where x s is the value of the variable of interest (e.g., time to cure, salary) of the member of the i th pair from one group (called the control group) and y t the corresponding value of the member of the pair from the second group (often called the treatment or study group). The difference

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Interpreting Statistics for Beginners

  A Guide for Behavioural and Social Scientists

  • Vladimir Hedrih, Andjelka Hedrih(Authors)
  • 2022(Publication Date)
  • Routledge

    (Publisher)

  smaller effect sizes to reject the null hypothesis or smaller sample sizes for the same effect size to reach statistical significance than tests for independent samples. In other words, they typically have higher power compared to tests created for independent samples.

  7.2 Comparing two means – t test

  The t test is a parametric test created for determining whether two samples come from populations with the same value of the arithmetic mean. This is done by testing the null hypothesis that the difference between means of two populations is zero. This is based on the calculation of a standard error of the differences of the mean and then dividing the obtained difference between sample means with the standard error of difference between sample means. As with all standard errors, standard error of difference between means then depends on the variability of the samples on which they were calculated (i.e. their standard deviations/variances) and on sample size. The larger the sample and the smaller the variability of values of entities in both samples, the smaller the standard error of difference between means will be. The difference between means is than calculated and divided by this standard error of differences between means and the result obtained in this way is called the t statistic. The t statistic is than used to estimate the probability of obtaining the difference between means as was obtained between these two samples or larger when the difference between population means is 0 and this probability is the statistical significance of the t test. If the result of the t test is statistically significant i.e. if p is lower than the accepted threshold of statistical significance (most commonly .05), we can reject the null hypothesis and conclude that samples do not come from populations with the same mean on the examined variable. On the other hand, if the value of statistical significance is above the accepted threshold, we accept the null hypothesis and conclude that there is insufficient evidence that population means are not the same (i.e. we will treat them as if they were the same).

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Essentials of Business Statistics

  • Ken Black, Ignacio Castillo, Amy Goldlist, Timothy Edmunds(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  Using this pooled estimate in the formula for the standard error of the difference gives us = − + − + − + s s n s n n n n n ( 1) ( 1) 2 1 1 d a a b b a b a b 2 2 This leads to the formula for the t statistic for the difference between two means. t Formula to Test the Difference in Means Assuming A σ and B σ Are Equal t d s a b s n s n n n n n ( ) ( ) ( 1) ( 1) 2 1 1 test d A B a a b b a b a b 2 2 δ µ µ = − = − − − − + − + − + n n Degrees of freedom 2 a b ν = = + − (12.3) Assumptions As usual, we have some assumptions. We are relying on the central limit theorem again, so we need either the samples to be large (n n 30, 30 a b > > ) or the populations to be normally distributed. Also, we are explicitly assuming that the two populations have (approximately) equal standard deviations (which is also often expressed as the populations having approxi- mately equal variances). POINTS OF INTEREST How close is close enough? If we do not know the population variances, it is hard to say whether they are the same. To this end, we will use the rough metric that we can use this test when the larger sample variance is no more than three times as large as the other. As with paired difference tests, a common test is to test whether δ = 0, that is, there is no difference between the two means. This type of analysis can be used to determine, for exam- ple, whether the effectiveness of two brands of toothpaste is different or whether two brands of tires wear differently. We can also study situations where we are interested in not just any difference between the population means, but whether that difference is larger (or smaller) Hypothesis Testing of Unpaired Means 403 than a specific size. For example, we might wonder whether the efficiency of a new machine is enough of an improvement over an older model to justify the expense of switching.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Fundamentals of Biostatistics

  • Bernard Rosner, , , (Authors)
  • 2015(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  If our two samples are paired—that is, if each person is used as his or her own control or if the samples consist of different people who are matched on a one-to-one basis—then the paired t test is appropriate. If the samples are independent, then the F test for the equality of two variances is used to decide whether the variances are significantly different. If the variances are not significantly different, then the two-sample t test with equal variances is used; if the variances are significantly different, then the two-sample t test with unequal variances is used. If we are only comparing the variances of the two samples, then only the F test for comparing variances is used, as indicated in the lower left of Figure 8.13. The chapter then provided methods for determining the appropriate sam-ple size and power formulas for planning investigations in which the goal is to compare the means from two independent samples and concluded with methods for detection of outliers. In Chapter 9, we extend our work on the comparison of two samples to the case in which there are two groups to be compared but the assumption of normality is questionable. We will introduce nonparametric methods to solve this problem to complement the parametric methods discussed in Chapters 7 and 8. P R O B L E M S 8.1 Find the lower 2.5th percentile of an F distribution with 14 and 7 df . What symbol is used to denote this? Nutrition The mean ± 1 sd of ln [calcium intake (mg)] among 25 females, 12 to 14 years of age, below the poverty level is 6.56 ± 0.64. Similarly, the mean ± 1 sd of ln [calcium intake (mg)] among 40 females, 12 to 14 years of age, above the poverty level is 6.80 ± 0.76. 8.2 Test for a significant difference between the variances of the two groups. 8.3 What is the appropriate procedure to test for a signifi-cant difference in means between the two groups? 8.4 Implement the procedure in Problem 8.3 using the critical-value method.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Statistical Concepts for the Behavioral Sciences

  • Harold O. Kiess, Bonnie A. Green(Authors)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  This decision implies that there is no evidence to reject the hypothesis that the difference between the population means is zero. The 95 percent confidence interval for the difference between these population means is to , where , , , and . Substituting these numeri- cal values, we obtain, to or, to . Notice that this interval does include the difference of zero between the population means, a value that would occur if the null hypothesis is true. Thus, the t test and the confidence intervals provide a consistent result that the sample means differ only by chance. + .5 - 0.1 (3.2 - 3.0) + 2.056(0.134), (3.2 - 3.0) - 2.056(0.134) t .05 = 2.056 s X 1 - X 2 = 0.134 X 2 = 3.0 X 1 = 3.2 (X 1 - X 2 ) + t .05 (s X 1 - X 2 ) (X 1 - X 2 ) - t .05 (s X 1 - X 2 )  1 Z  2  1 =  2  1 Z  2  1 =  2  1 =  2  1 -  2 1.2 + 1.0, 1.2 - 1.0 (7.8 - 6.6) + (2.074) (0.492), (7.8 - 6.6) - (2.074) (0.492)  1 -  2 0.492 s X 1 - X 2 = X 2 = 6.6 X 1 = 7.8 218 STATISTICAL CONCEPTS FOR THE BEHAVIORAL SCIENCES What Does a t Test Actually Test? The t test provides information needed for deciding whether a research hypothesis is or is not supported by the data of the experiment. It is important to recognize, however, that a research hypothesis is not tested directly by the t test. The t test is simply a procedure for testing the statistical hypothesis, H 0 : . Regardless of the independent and depen- dent variables stated in the research hypothesis, the null and alternative hypotheses for a two-tailed t test are always H 0 : and H 1 : , respectively. A research hypoth- esis for an experiment, however, is a statement predicting a relationship between an inde- pendent variable and a dependent variable. For example, the statement “it is expected that the number of anagram solutions for students given a red cover will be less than the num- ber of solutions of students given a green cover” is a research hypothesis.

  Sign up to read

  Learn more about book