Differences Between Groups - Using the t-Test

If you have a two-group research study and your groups are fairly large (20 to 100) then you can use the independent means t-test. The t-test is called a parametric test because your data must come from populations that are normally distributed and use interval measurement. The t-test is used to answer to this question: Is there any difference between the means of the two populations of which our data is a random sample? The t-test is also called a test of inference because we are trying to discover if populations are different by studying samples from the populations, i.e., what we find to be true about our samples we will assume to be true about the population.

To compute a t-test we need to obtain the following -

 Note: Y_1i and Y_2j stand for each individual score of the variables

1. The standard deviation of the difference between the means is:

   

 

 

2. The difference between the means is:

   

    

Note: If mean Y₁ is larger than mean Y₂ then the numerator will be negative, and thus the t-value will be negative.

 

3. The t-value is:

 

If the difference between the means is large in comparison to the standard deviation of the difference between the means, then the t-value is large. The larger the t-value the smaller the probability that the means of the two populations are the same. It does not matter if the t-value is negative or positive.  Use the absolute value (disregard the sign) when interpreting the t-value.

Let's look at an example: Suppose we are studying energy expenditure during ambulation (walking) with ortho crutches and axillary crutches. As a measure of energy expenditure we are using heart rate in beats per minute. Two groups of normal subjects (20 per group) walk at their own pace using crutches for 11.5 minutes. Heart rate is measured at the end of the 11.5 minutes. Group A uses the axillary crutches. Our hypothesis is that mean heart rate of Group B will be higher than Group A. We are assuming that it requires more energy to use the axillary crutch than the ortho crutch.

Using the formula we will solve for the t-value using the data above:

1.	Y1 = SY1i  =  2673  = 133.65      ───   ───       n1      20
2.	Y2 = SY1j  =  2778  = 138.90      ───   ───       n1      20
3.	SS1 = S(Y1i-Y1)2 = 846.55
4.	SS2 = S(Y2j-Y2)2 = 1477.80
	The standard deviations for each group are obtained by dividing SS1 by 19 (n-1) and SS2 by 19 and then taking the square roots. The standard deviation for Group A (Y1) = 6.67 and for Group B (Y2) = 8.82.
5.	= 2.47
6.	= 2.13

What does a t-value of 2.13 mean? The t-value is an indication of the probability that both populations from which we selected our samples have the same mean and that differences in our sample means are due to random fluctuation. As the t-value gets smaller (approaches zero) the probability that the population means are the same gets larger. As the t-value gets larger (in either the positive or negative direction) the probability that the population means are the same gets smaller.

We can use the t-value to decide between our two statistical hypotheses:

1. Null hypothesis : The two populations have the same mean

2. Alternative hypothesis : The population that uses the axillary crutch has a higher heart rate than the ortho crutch.

If our computed t-value is the same as or smaller than the tabled t-value, we accept the null hypothesis and conclude that the populations have the same mean. If our t-value is larger, we can accept the alternative hypothesis. Since our t-value (t = 2.13) is larger than the tabled t-value (t = 1.684) this means that there is a small chance (1 in 20) that the population means are the same, and so it is reasonable to conclude that the means are different.

Abridged from Fisher and Yates. Statistical
Tables for Biological, Agricultural, and Medical
Research. Edinburgh, Oliver and Boyd Limited.

Another t-test is available when you have one-group and observe the continuous, dependent variable twice. This is the dependent means t-test (or matched pairs t-test). The word dependent means that the second observation is related to the first since the same group is being measured twice. This is different from the independent means t-test where two different measured once. Use the equation below to compute the t-value. Interpretation is the same as for the independent means t-test.

Where, D is the mean difference between the two observations

S_d is the standard deviation of the differences, and

n is the number of subjects.

Statistical Tests and Probability

Suppose we draw random samples from a population and study the average value of a variable. We would find that most of the time the means of those samples would be almost the same, and sometimes they would be very different. This difference would be a random event. When we do research we expect the independent variable to cause the mean of the dependent variable to be different for each group. A statistical test tells us the probability that the difference we find is due to a random event rather than due to the independent variable.

The graph below shows the distribution of 80,788 t-tests run on 80,788 pairs of groups of twenty random numbers. The t-test values from our random distribution range from -3 to +3 and indicate the size of the difference between the two groups. The majority of the t-test values are near zero.

By counting the number of t-values in the random distribution that are larger than a particular t-value, and dividing by the total number of t-values (80,788), we can compute the probability that our experimental t-value was due to chance. The table below does this when t = 1.7 and 2.5:

The various tables of critical values (t-test, correlation, chi-square) are summaries of many distributions like our t-value random distribution for particular probability levels (.10, .05, and .01).