We let R 1 denote the sum of the ranks in group 1 i. If the null hypothesis is true i. In this example, the lower values lower ranks are clustered in the new drug group group 2 , while the higher values higher ranks are clustered in the placebo group group 1. This is suggestive, but is the observed difference in the sums of the ranks simply due to chance?
To answer this we will compute a test statistic to summarize the sample information and look up the corresponding value in a probability distribution.
Is this evidence in support of the null or research hypothesis? Before we address this question, we consider the range of the test statistic U in two different situations. Consider the situation where there is complete separation of the groups, supporting the research hypothesis that the two populations are not equal.
If all of the higher numbers of episodes of shortness of breath and thus all of the higher ranks are in the placebo group, and all of the lower numbers of episodes and ranks are in the new drug group and that there are no ties, then:. Consider a second situation where l ow and high scores are approximately evenly distributed in the two groups , supporting the null hypothesis that the groups are equal.
If ranks of 2, 4, 6, 8 and 10 are assigned to the numbers of episodes of shortness of breath reported in the placebo group and ranks of 1, 3, 5, 7 and 9 are assigned to the numbers of episodes of shortness of breath reported in the new drug group, then:. Thus, smaller values of U support the research hypothesis, and larger values of U support the null hypothesis. In the example above, U can range from 0 to 25 and smaller values of U support the research hypothesis i.
The procedure for determining exactly when to reject H 0 is described below. In every test, we must determine whether the observed U supports the null or research hypothesis. This is done following the same approach used in parametric testing. Specifically, we determine a critical value of U such that if the observed value of U is less than or equal to the critical value, we reject H 0 in favor of H 1 and if the observed value of U exceeds the critical value we do not reject H 0.
The critical value of U can be found in the table below. However, in this example, the failure to reach statistical significance may be due to low power. The sample data suggest a difference, but the sample sizes are too small to conclude that there is a statistically significant difference.
A new approach to prenatal care is proposed for pregnant women living in a rural community. The new program involves in-home visits during the course of pregnancy in addition to the usual or regularly scheduled visits.
A pilot randomized trial with 15 pregnant women is designed to evaluate whether women who participate in the program deliver healthier babies than women receiving usual care. Each of the 5 criteria is rated as 0 very unhealthy , 1 or 2 healthy based on specific clinical criteria. Infants with scores of 7 or higher are considered normal, low and 0 to 3 critically low.
Sometimes the APGAR scores are repeated, for example at 1 minute after birth, at 5 and at 10 minutes after birth and analyzed. Virginia Apgar and is used to describe the condition of an infant at birth. Recall that APGAR scores range from 0 to 10 with scores of 7 or higher considered normal healthy , low and critically low. Is there statistical evidence of a difference in APGAR scores in women receiving the new and enhanced versus usual prenatal care? We run the test using the five-step approach.
H 1 : The two populations are not equal. The test statistic is U, the smaller of. The appropriate critical value can be found in the table above. The first step is to assign ranks of 1 through 15 to the smallest through largest values in the total sample, as follows:. Next, we sum the ranks in each group. We reject H 0 because 9.
A clinical trial is run to assess the effectiveness of a new anti-retroviral therapy for patients with HIV. Patients are randomized to receive a standard anti-retroviral therapy usual care or the new anti-retroviral therapy and are monitored for 3 months. The primary outcome is viral load which represents the number of HIV copies per milliliter of blood. A total of 30 participants are randomized and the data are shown below.
Is there statistical evidence of a difference in viral load in patients receiving the standard versus the new anti-retroviral therapy? Because viral load measures are not normally distributed with outliers as well as limits of detection e. The first step is to assign ranks of 1 through 30 to the smallest through largest values in the total sample. Note in the table below, that the "undetectable" measurement is listed first in the ordered values smallest and assigned a rank of 1.
We now compute U 1 and U 2 , as follows,. Therefore, a researcher decided to investigate whether an exercise or weight loss intervention was more effective in lowering cholesterol levels. To this end, the researcher recruited a random sample of inactive males that were classified as overweight. This sample was then randomly split into two groups: Group 1 underwent a calorie-controlled diet i.
In order to determine which treatment programme was more effective, cholesterol concentrations were compared between the two groups at the end of the treatment programmes.
In SPSS Statistics, we entered the scores for cholesterol concentration, our dependent variable, under the variable name Cholesterol. Next, we created a grouping variable, called Group , which represented our independent variable. Since our independent variable had two groups - 'diet' and 'exercise' - we gave the diet group a value of "1" and the exercise group a value of "2".
How we have explained the data setup above relates to both procedures when your dependent variable is continuous , which is what we take you through in the Test Procedure in SPSS Statistics section next. We explain how to do this in our enhanced Mann-Whitney U test guide, which you can access by subscribing to Laerd Statistics.
If you read assumption 4 earlier, you'll know that the SPSS Statistics procedure when analysing your data using a Mann-Whitney U test is different depending on the shape of the two distributions of your independent variable.
In our example, where our dependent variable is cholesterol concentration, Cholesterol , we are referring to the two distributions of the independent variable, Group i.
In the 10 steps below, we show you how to analyse your data using a Mann-Whitney U test in SPSS Statistics when these two distributions have a different shape , and therefore, you have to compare the mean ranks of your dependent variable rather than medians. To use SPSS Statistics to determine whether your two distributions have the same or different shapes , or if you want to know how to use SPSS Statistics to carry out a Mann-Whitney U test when your two distributions have the same shape , such that you need to compare medians rather than mean ranks , you will need to access the Procedures section of our enhanced Mann-Whitney U test guide N.
We show you both procedures in our enhanced Mann-Whitney U test guide. At the end of the 9 steps below, we show you how to interpret the results from this test using mean ranks. However, the procedure is identical. Note: Make sure that the M ann-Whitney U checkbox is ticked in the —Test Type— area and the G rouping Variable: box is highlighted in yellow as seen above.
If it is not highlighted in yellow, simply click your cursor in the G rouping Variable: box to highlight it. Click through to the next page for the remaining procedure and how to interpret the output. SPSS Statistics Assumptions When you choose to analyse your data using a Mann-Whitney U test, part of the process involves checking to make sure that the data you want to analyse can actually be analysed using a Mann-Whitney U test. Examples of ordinal variables include Likert items e.
Examples of continuous variables include revision time measured in hours , intelligence measured using IQ score , exam performance measured from 0 to , weight measured in kg , and so forth. You can learn more about ordinal and continuous variables in our article: Types of Variable.
Assumption 2: Your independent variable should consist of two categorical , independent groups. Example independent variables that meet this criterion include gender 2 groups: male or female , employment status 2 groups: employed or unemployed , smoker 2 groups: yes or no , and so forth.
Assumption 3: You should have independence of observations , which means that there is no relationship between the observations in each group or between the groups themselves. For example, there must be different participants in each group with no participant being in more than one group. This is more of a study design issue than something you can test for, but it is an important assumption of the Mann-Whitney U test.
If your study fails this assumption, you will need to use another statistical test instead of the Mann-Whitney U test e. If you are unsure whether your study meets this assumption, you can use our Statistical Test Selector , which is part of our enhanced content.
Assumption 4: A Mann-Whitney U test can be used when your two variables are not normally distributed. However, in order to know how to interpret the results from a Mann-Whitney U test, you have to determine whether your two distributions i.
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