Which t test to use

Degrees of freedom refers to the values in a study that has the freedom to vary and are essential for assessing the importance and the validity of the null hypothesis. Computation of these values usually depends upon the number of data records available in the sample set. The correlated t-test is performed when the samples typically consist of matched pairs of similar units, or when there are cases of repeated measures.

For example, there may be instances of the same patients being tested repeatedly—before and after receiving a particular treatment. In such cases, each patient is being used as a control sample against themselves.

This method also applies to cases where the samples are related in some manner or have matching characteristics, like a comparative analysis involving children, parents or siblings.

Correlated or paired t-tests are of a dependent type, as these involve cases where the two sets of samples are related. The formula for computing the t-value and degrees of freedom for a paired t-test is:. The remaining two types belong to the independent t-tests. They include cases like a group of patients being split into two sets of 50 patients each. One of the groups becomes the control group and is given a placebo, while the other group receives the prescribed treatment.

This constitutes two independent sample groups which are unpaired with each other. The equal variance t-test is used when the number of samples in each group is the same, or the variance of the two data sets is similar. The following formula is used for calculating t-value and degrees of freedom for equal variance t-test:. The unequal variance t-test is used when the number of samples in each group is different, and the variance of the two data sets is also different.

This test is also called the Welch's t-test. The following formula is used for calculating t-value and degrees of freedom for an unequal variance t-test:. The following flowchart can be used to determine which t-test should be used based on the characteristics of the sample sets. The key items to be considered include whether the sample records are similar, the number of data records in each sample set, and the variance of each sample set.

Assume that we are taking a diagonal measurement of paintings received in an art gallery. One group of samples includes 10 paintings, while the other includes 20 paintings. The data sets, with the corresponding mean and variance values, are as follows:.

Though the mean of Set 2 is higher than that of Set 1, we cannot conclude that the population corresponding to Set 2 has a higher mean than the population corresponding to Set 1. Is the difference from We establish the problem by assuming the null hypothesis that the mean is the same between the two sample sets and conduct a t-test to test if the hypothesis is plausible.

The t-value is Since the minus sign can be ignored when comparing the two t-values, the computed value is 2. The degrees of freedom value is One can specify a level of probability alpha level, level of significance, p as a criterion for acceptance.

Comparing this value against the computed value of 2. Therefore, it is safe to reject the null hypothesis that there is no difference between means. The population set has intrinsic differences, and they are not by chance. What is your plagiarism score? Compare your paper with over 60 billion web pages and 30 million publications.

From the output table, we can see that the difference in means for our sample data is Our p -value of 2. What is a t-test? What does a t-test measure? Which t-test should I use? What is the difference between a one-sample t-test and a paired t-test? Can I use a t-test to measure the difference among several groups? Is this article helpful? Rebecca Bevans Rebecca is working on her PhD in soil ecology and spends her free time writing.

She's very happy to be able to nerd out about statistics with all of you. Other students also liked. Statistical tests: which one should you use? Your choice of statistical test depends on the types of variables you're dealing with and whether your data meets certain assumptions. A step-by-step guide to hypothesis testing Hypothesis testing is a formal procedure for investigating our ideas about the world. It allows you to statistically test your predictions.

Test statistics explained The test statistic is a number, calculated from a statistical test, used to find if your data could have occurred under the null hypothesis. One final method for comparing distributions is worth mentioning. We noted previously that one of the assumptions for the t-test is that the variances of the two samples are equal. However, a modification of the t-test known as Welch's test is said to correct for this problem by estimating the variances, and adjusting the degrees of freedom to use in the test.

This correction is performed by default, but can be shut off by using the var. Let's see how it works:. Since the statistic is the same in both cases, it doesn't matter whether we use the correction or not; either way we'll see identical results when we compare the two methods using the techniques we've already described.

Since the degree of freedom correction changes depending on the data, we can't simply perform the simulation and compare it to a different number of degrees of freedom. The other thing that changes when we apply the correction is the p-value that we would use to decide if there's enough evidence to reject the null hypothesis. What is the behaviour of the p-values? While not necessarily immediately obvious, under the null hypothesis, the p-values for any statistical test should form a uniform distribution between 0 and 1; that is, any value in the interval 0 to 1 is just as likely to occur as any other value.

For a uniform distribution, the quantile function is just the identity function. A value of. As a quick check of this notion, let's look at the density of probability values when the null hypothesis is true:.

Another way to check to see if the probabilities follow a uniform distribution is with a QQ plot:. Now, let's look at some of the quantiles of the p-values when we force the t.

The agreement actually looks very good. What about when we let t. There's not that much of a difference, but, of course, the variances in this example were equal. How does the correction work when the variances are not equal? Significant differences among group means are calculated using the F statistic, which is the ratio of the mean sum of squares the variance explained by the independent variable to the mean square error the variance left over. If the F statistic is higher than the critical value the value of F that corresponds with your alpha value, usually 0.

If you are only testing for a difference between two groups, use a t-test instead. The formula for the test statistic depends on the statistical test being used. Generally, the test statistic is calculated as the pattern in your data i. Linear regression most often uses mean-square error MSE to calculate the error of the model. MSE is calculated by:. Linear regression fits a line to the data by finding the regression coefficient that results in the smallest MSE. Simple linear regression is a regression model that estimates the relationship between one independent variable and one dependent variable using a straight line.

Both variables should be quantitative. For example, the relationship between temperature and the expansion of mercury in a thermometer can be modeled using a straight line: as temperature increases, the mercury expands. This linear relationship is so certain that we can use mercury thermometers to measure temperature.

A regression model is a statistical model that estimates the relationship between one dependent variable and one or more independent variables using a line or a plane in the case of two or more independent variables. A regression model can be used when the dependent variable is quantitative, except in the case of logistic regression, where the dependent variable is binary.

A t-test should not be used to measure differences among more than two groups, because the error structure for a t-test will underestimate the actual error when many groups are being compared.

A one-sample t-test is used to compare a single population to a standard value for example, to determine whether the average lifespan of a specific town is different from the country average. A paired t-test is used to compare a single population before and after some experimental intervention or at two different points in time for example, measuring student performance on a test before and after being taught the material.

A t-test measures the difference in group means divided by the pooled standard error of the two group means. In this way, it calculates a number the t-value illustrating the magnitude of the difference between the two group means being compared, and estimates the likelihood that this difference exists purely by chance p-value.

A t-test is a statistical test that compares the means of two samples. It is used in hypothesis testing , with a null hypothesis that the difference in group means is zero and an alternate hypothesis that the difference in group means is different from zero. Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test.

Significance is usually denoted by a p -value , or probability value. Statistical significance is arbitrary — it depends on the threshold, or alpha value, chosen by the researcher. When the p -value falls below the chosen alpha value, then we say the result of the test is statistically significant.

A test statistic is a number calculated by a statistical test. It describes how far your observed data is from the null hypothesis of no relationship between variables or no difference among sample groups.

The test statistic tells you how different two or more groups are from the overall population mean , or how different a linear slope is from the slope predicted by a null hypothesis. Different test statistics are used in different statistical tests. The measures of central tendency you can use depends on the level of measurement of your data.

Ordinal data has two characteristics:. Nominal and ordinal are two of the four levels of measurement. Nominal level data can only be classified, while ordinal level data can be classified and ordered. If your confidence interval for a difference between groups includes zero, that means that if you run your experiment again you have a good chance of finding no difference between groups.

If your confidence interval for a correlation or regression includes zero, that means that if you run your experiment again there is a good chance of finding no correlation in your data. In both of these cases, you will also find a high p -value when you run your statistical test, meaning that your results could have occurred under the null hypothesis of no relationship between variables or no difference between groups.

If you want to calculate a confidence interval around the mean of data that is not normally distributed , you have two choices:. The standard normal distribution , also called the z -distribution, is a special normal distribution where the mean is 0 and the standard deviation is 1. Any normal distribution can be converted into the standard normal distribution by turning the individual values into z -scores. In a z -distribution, z -scores tell you how many standard deviations away from the mean each value lies.

The z -score and t -score aka z -value and t -value show how many standard deviations away from the mean of the distribution you are, assuming your data follow a z -distribution or a t -distribution. These scores are used in statistical tests to show how far from the mean of the predicted distribution your statistical estimate is.

If your test produces a z -score of 2. The predicted mean and distribution of your estimate are generated by the null hypothesis of the statistical test you are using. The more standard deviations away from the predicted mean your estimate is, the less likely it is that the estimate could have occurred under the null hypothesis.

To calculate the confidence interval , you need to know:. Then you can plug these components into the confidence interval formula that corresponds to your data. The formula depends on the type of estimate e. The confidence level is the percentage of times you expect to get close to the same estimate if you run your experiment again or resample the population in the same way. The confidence interval is the actual upper and lower bounds of the estimate you expect to find at a given level of confidence.

These are the upper and lower bounds of the confidence interval. Nominal data is data that can be labelled or classified into mutually exclusive categories within a variable. These categories cannot be ordered in a meaningful way. For example, for the nominal variable of preferred mode of transportation, you may have the categories of car, bus, train, tram or bicycle.

The mean is the most frequently used measure of central tendency because it uses all values in the data set to give you an average. Statistical tests commonly assume that:. If your data does not meet these assumptions you might still be able to use a nonparametric statistical test , which have fewer requirements but also make weaker inferences. Measures of central tendency help you find the middle, or the average, of a data set.

Some variables have fixed levels. For example, gender and ethnicity are always nominal level data because they cannot be ranked. However, for other variables, you can choose the level of measurement. For example, income is a variable that can be recorded on an ordinal or a ratio scale:.

If you have a choice, the ratio level is always preferable because you can analyze data in more ways. The higher the level of measurement, the more precise your data is. The level at which you measure a variable determines how you can analyze your data. Depending on the level of measurement , you can perform different descriptive statistics to get an overall summary of your data and inferential statistics to see if your results support or refute your hypothesis. Levels of measurement tell you how precisely variables are recorded.

There are 4 levels of measurement, which can be ranked from low to high:. The p -value only tells you how likely the data you have observed is to have occurred under the null hypothesis. The alpha value, or the threshold for statistical significance , is arbitrary — which value you use depends on your field of study.

In most cases, researchers use an alpha of 0. P -values are usually automatically calculated by the program you use to perform your statistical test.

They can also be estimated using p -value tables for the relevant test statistic. P -values are calculated from the null distribution of the test statistic. They tell you how often a test statistic is expected to occur under the null hypothesis of the statistical test, based on where it falls in the null distribution. If the test statistic is far from the mean of the null distribution, then the p -value will be small, showing that the test statistic is not likely to have occurred under the null hypothesis.

A p -value , or probability value, is a number describing how likely it is that your data would have occurred under the null hypothesis of your statistical test. You can choose the right statistical test by looking at what type of data you have collected and what type of relationship you want to test.

The test statistic will change based on the number of observations in your data, how variable your observations are, and how strong the underlying patterns in the data are. For example, if one data set has higher variability while another has lower variability, the first data set will produce a test statistic closer to the null hypothesis, even if the true correlation between two variables is the same in either data set.

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