Tukey post hoc test

A hospital wants to know how a homeopathic medicine for depression performs in comparison to alternatives.

Learn How to Calculate Tukey's Post HOC Test - Tutorial

They adminstered 4 treatments to patients for 2 weeks and then measured their depression levels. The data, part of which are shown above, are in depression. Before running any statistical test, always make sure your data make sense in the first place.

In this case, a split histogram basically tells the whole story in a single chart. We don't see many SPSS users run such charts but you'll see in a minute how incredibly useful it is. The screenshots below show how to create it. In step below, you can add a nice title to your chart. Clicking P aste results in the syntax below.

Running it creates our chart. We'll now take a more precise look at our data by running a means table. We could do so from A nalyze C ompare Means M eans but the syntax is so simple that just typing it is probably faster. Unsurprisingly, our table mostly confirms what we already saw in our histogram.

Well, for our sample we can. For our population all people suffering from depression we can't. The basic problem here is that samples differ from the populations from which they are drawn. If our four medicines perform equally well in our population, then we may still see some differences between our sample means.

However, large sample differences are unlikely if all medicines perform equally in our population. The question we'll now answer is: are the sample means different enough to reject the null hypothesis that the mean BDI scores in our populations are all equal?

However, it could be argued that you should always run post hoc tests. In some fields like market research, this is pretty common. Reversely, you could argue that you should never use post hoc tests because the omnibus test suffices: some analysts claim that running post hoc tests is overanalyzing the data.

Many social scientists are completely obsessed with statistical significance -because they don't understand what it really means- and neglect what's more interesting: effect sizes and confidence intervals. In any case, the idea of post hoc tests is clarified best by just running them.Since these are independent and not paired or correlated, the number of observations of each treatment may be different.

This calculator is hard-coded for a maximum of 10 treatments, which is more than adequate for most researchers. But it stops there in its tracks. This self-contained calculator, with flexibility to vary the number of treatments columns to be compared, starts with one-way ANOVA. However, it lacks the key built-in statistical function needed for conducting Excel-contained Tukey HSD. Continuing education in Statistics The hard-core statistical packages demand a certain expertise to format the input data, write code to implement the procedures and then decipher their s Old School Mainframe Era output.

This is the right tool for you!

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It was inspired by the frustration of several biomedical scientists with learning the software setup and coding of these serious statistical packages, almost like operating heavy bulldozer machinery to swat an irritating mosquito.

For code grandmasters, fully working code and setup instructions are provided for replication of the results in the serious academic-research-grade open-source and hence free R statistical package. Tukey originated his HSD test, constructed for pairs with equal number of samples in each treatment, way back in When the sample sizes are unequal, we the calculator automatically applies the Tukey-Kramer method Kramer originated in A decent writeup on these relevant formulae appear in the Tukey range test Wiki entry.

The NIST Handbook page mentions this modification but dooes not provide the formula, while the Wiki entry makes adequately specifies it.

However, this calculator is hard-coded for contrasts that are pairsand hence does not pester the user for additional input that defines generalized contrast structures. The Bonferroni and Holm methods of multiple comparison depends on the number of relevant pairs being compared simultaneously. This calculator is hard-coded for Bonferroni and Holm simultaneous multiple comparison of 1 all pairs and 2 only a subset of pairs relative to one treatment, the first column, deemed to be the control.

The post-hoc Bonferroni simultaneous multiple comparison of treatment pairs by this calculator is based on the formulae and procedures at the NIST Engineering Statistics Handbook page on Bonferroni's method. The original Bonferroni published paper in Italian dating back to is hard to find on the web. A significant improvement over the Bonferroni method was proposed by Holm Among the many reviews of the merits of the Holm method and its uniform superiority over the Bonferroni method, that of Aickin and Gensler is notable.

This paper is the also source of our algorithm to make comparisons according to the Holm method. All statistical packages today incorporate the Holm method. There is wide agreement that each of these three methods have their merits. The recommendation on the relative merits and advantages of each of these methods in the NIST Engineering Statistics Handbook page on comparison of these methods are reproduced below:.

The following excerpts from Aickin and Gensler makes it clear that the Holm method is uniformly superior to the Bonferroni method:. If only a subset of pairwise comparisons are required, Bonferroni may sometimes be better. Many computer packages include all three methods. So, study the output and select the method with the smallest confidence band.

No single method of multiple comparisons is uniformly best among all the methods.The idea behind the Tukey HSD Honestly Significant Difference test is to focus on the largest value of the difference between two group means.

tukey post hoc test

The relevant statistic is. The statistic q has a distribution called the studentized range q see Studentized Range Distribution. Thus we can use the following t statistic. From these observations we can calculate confidence intervals in the usual way:. Since the difference between the means for women taking the drug and women in the control group is 5. The following table shows the same comparisons for all pairs of variables:. From Figure 1 we see that the only significant difference in means is between women taking the drug and men in the control group i.

In Figure 2 we compute the confidence interval for the comparison requested in the example as well as for the variables with maximum difference. These function are based on the table of critical values provided in Studentized Range q Table. The Real Statistics Resource Pack also provides the following functions which provide estimates for the Studentized range distribution and its inverse based on a somewhat complicated algorithm.

QDIST 4. To get the usual cdf value for the Studentized range distribution, you need to divide the result from QDIST by 2, which for this example is.

C n ,2 rows if the data in R1 contains n columns. The first two columns contain the column numbers in R1 from 1 to n that are being compared and the third column contains the p-values for each of the pairwise comparisons.

RSS - Posts. RSS - Comments. Real Statistics Using Excel. Everything you need to perform real statistical analysis using Excel. Skip to content. The critical value for differences in means is Since the difference between the means for women taking the drug and women in the control group is 5. Real Statistics Resources. Follow Real1Statistics. Search for:.

Charles Zaiontz.

tukey post hoc test

Proudly powered by WordPress.The Tukey HSD "honestly significant difference" or "honest significant difference" test is a statistical tool used to determine if the relationship between two sets of data is statistically significant — that is, whether there's a strong chance that an observed numerical change in one value is causally related to an observed change in another value. In other words, the Tukey test is a way to test an experimental hypothesis.

The Tukey test is invoked when you need to determine if the interaction among three or more variables is mutually statistically significant, which unfortunately is not simply a sum or product of the individual levels of significance. Simple statistics problems involve looking at the effects of one independent variable, like the number of hours studied by each student in a class for a particular test, on a second dependent variable, like the student's scores on the test.

Then you refer to a t-table that takes into account the number of data pairs in your experiment to see if your hypothesis was correct. Sometimes, however, the experiment may look at multiple independent or dependent variables simultaneously. For example, in the above example, the hours of sleep each student got the night before the test and his or her class grade going in might be included.

Such multivariate problems require something other than a t-test owing to the sheer number if independently varying relationships. ANOVA stands for "analysis of variance" and addresses precisely the problem just described. It accounts for the rapidly expanding degrees of freedom in a sample as variables are added.

For example, looking at hours vs. In an ANOVA test, the variable of interest after calculations have been run is F, which is the found variation of the averages of all of the pairs, or groups, divided by the expected variation of these averages.

The higher this number, the stronger the relationship, and "significance" is usually set at 0. John Tukey came up with the test that bears his name when he realized the mathematical pitfalls of trying to use independent P-values to determine the utility of a multiple-variables hypothesis as a whole. At the time, t-tests were being applied to three or more groups, and he considered this dishonest — hence "honestly significant difference. What his test does is compare the differences between means of values rather than comparing pairs of values.

The value of the Tukey test is given by taking the absolute value of the difference between pairs of means and dividing it by the standard error of the mean SE as determined by a one-way ANOVA test. The SE is in turn the square root of variance divided by sample size.

An example of an online calculator is listed in the Resources section. The Tukey test is a post hoc test in that the comparisons between variables are made after the data has already been collected.

This differs from an a priori test, in which these comparisons are made in advance. In the former case, you might look at the mile run times of students in three different phys-ed classes one year. In the latter case, you might assign students to one of three teachers and then have them run a timed mile.

Kevin Beck holds a bachelor's degree in physics with minors in math and chemistry from the University of Vermont. Formerly with ScienceBlogs. More about Kevin and links to his professional work can be found at www. About the Author. Copyright Leaf Group Ltd.An ANOVA is a statistical test that is used to determine whether or not there is a statistically significant difference between the means of three or more independent groups. The alternative hypothesis: Ha : at least one of the means is different from the others.

It simply tells us that not all of the group means are equal. If the p-value is not statistically significant, this indicates that the means for all of the groups are not different from each other, so there is no need to conduct a post hoc test to find out which groups are different from each other.

As mentioned before, post hoc tests allow us to test for difference between multiple group means while also controlling for the family-wise error rate. In a hypothesis testthere is always a type I error rate, which is defined by our significance level alpha and tells us the probability of rejecting a null hypothesis that is actually true. When we perform one hypothesis test, the type I error rate is equal to the significance level, which is commonly chosen to be 0.

However, when we conduct multiple hypothesis tests at once, the probability of getting a false positive increases. For example, imagine that we roll a sided dice.

Tukey's Post HOC Test Calculator

If we roll five dice at once, the probability increases to For example, suppose we have four groups: A, B, C, and D. This means there are a total of six pairwise comparisons we want to look at with a post hoc test:.

If we have more than four groups, the number of pairwise comparisons we will want to look at will only increase even more. The following table illustrates how many pairwise comparisons are associated with each number of groups along with the family-wise error rate:. Notice that the family-wise error rate increases rapidly as the number of groups and consequently the number of pairwise comparisons increases.

This means we would have serious doubts about our results if we were to make this many pairwise comparisons, knowing that our family-wise error rate was so high.

Fortunately, post hoc tests provide us with a way to make multiple comparisons between groups while controlling the family-wise error rate. This means we have sufficient evidence to reject the null hypothesis that all of the group means are equal.

Next, we can use a post hoc test to find which group means are different from each other. We will walk through examples of the following post hoc tests:.

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R gives us two metrics to compare each pairwise difference:. Both the confidence interval and the p-value will lead to the same conclusion. In particular, we know that the difference is positive, since the lower bound of the confidence interval is greater than zero.Although ANOVA is a powerful and useful parametric approach to analyzing approximately normally distributed data with more than two groups referred to as 'treatments'it does not provide any deeper insights into patterns or comparisons between specific groups.

After a multivariate test, it is often desired to know more about the specific groups to find out if they are significantly different or similar. This step after analysis is referred to as 'post-hoc analysis' and is a major step in hypothesis testing. One common and popular method of post-hoc analysis is Tukey's Test. The test is known by several different names.

Tukey's test compares the means of all treatments to the mean of every other treatment and is considered the best available method in cases when confidence intervals are desired or if sample sizes are unequal Wikipedia.

The outputs from two different but similar implementations of Tukey's Test will be examined along with how to manually calculate the test. Other methods of post-hoc analysis will be explored in future posts. ANOVA in this example is done using the aov function. The summary of the aov output is the same as the output of the anova function that was used in the previous example. To investigate more into the differences between all groups, Tukey's Test is performed.

The output gives the difference in means, confidence levels and the adjusted p-values for all possible pairs. The confidence levels and p-values show the only significant between-group difference is for treatments 1 and 2.

Note the other two pairs contain 0 in the confidence intervals and thus, have no significant difference. The results can also be plotted. Another way of performing Tukey's Test is provided by the agricolae package.

The HSD. The results from both tests can be verified manually.

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We'll start with the latter test HSD. The MSE calculation is the same as the previous example. With the q-value found, the Honestly Significant Difference can be determined. The Honestly Significant Difference is defined as the q-value multiplied by the square root of the MSE divided by the sample size. As mentioned earlier, the Honestly Significant Difference is a statistic that can be used to determine significant differences between groups. If the absolute value of the difference of the two groups' means is greater than or equal to the HSD, the difference is significant.

The means of each group can be found using the tapply function. Since there's only three groups, I went ahead and just calculated the differences manually.

tukey post hoc test

With the differences obtained, compare the absolute value of the difference to the HSD. I used a quick and dirty for loop to do this.

The output of the for loop shows the only significant difference higher than the HSD is between treatment 1 and 2.

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Since the test uses the studentized range, estimation is similar to the t-test setting. The Tukey-Kramer method allows for unequal sample sizes between the treatments and is, therefore, more often applicable though it doesn't matter in this case since the sample sizes are equal.

The Tukey-Kramer method is defined as:. Entering the values that were found earlier into the equation yields the same intervals as was found from the TukeyHSD output. The table from the TukeyHSD output is reconstructed below. Adjusted p-values are left out intentionally.Post hoc multiple comparison tests. Once you have determined that differences exist among the means, post hoc range tests and pairwise multiple comparisons can determine which means differ.

Comparisons are made on unadjusted values. These tests are used for fixed between-subjects factors only.

Performing a One-way ANOVA in Excel with post-hoc t-tests

In GLM Repeated Measures, these tests are not available if there are no between-subjects factors, and the post hoc multiple comparison tests are performed for the average across the levels of the within-subjects factors. For GLM Multivariate, the post hoc tests are performed for each dependent variable separately. The Bonferroni and Tukey's honestly significant difference tests are commonly used multiple comparison tests.

The Bonferroni testbased on Student's t statistic, adjusts the observed significance level for the fact that multiple comparisons are made. Sidak's t test also adjusts the significance level and provides tighter bounds than the Bonferroni test.

Tukey's honestly significant difference test uses the Studentized range statistic to make all pairwise comparisons between groups and sets the experimentwise error rate to the error rate for the collection for all pairwise comparisons. When testing a large number of pairs of means, Tukey's honestly significant difference test is more powerful than the Bonferroni test.

For a small number of pairs, Bonferroni is more powerful. Hochberg's GT2 is similar to Tukey's honestly significant difference test, but the Studentized maximum modulus is used. Usually, Tukey's test is more powerful. Gabriel's pairwise comparisons test also uses the Studentized maximum modulus and is generally more powerful than Hochberg's GT2 when the cell sizes are unequal. Gabriel's test may become liberal when the cell sizes vary greatly.

Dunnett's pairwise multiple comparison t test compares a set of treatments against a single control mean. The last category is the default control category. Alternatively, you can choose the first category. You can also choose a two-sided or one-sided test. To test that the mean at any level except the control category of the factor is not equal to that of the control category, use a two-sided test.

Multiple step-down procedures first test whether all means are equal. If all means are not equal, subsets of means are tested for equality.

These tests are more powerful than Duncan's multiple range test and Student-Newman-Keuls which are also multiple step-down proceduresbut they are not recommended for unequal cell sizes.

When the variances are unequal, use Tamhane's T2 conservative pairwise comparisons test based on a t testDunnett's T3 pairwise comparison test based on the Studentized maximum modulusGames-Howell pairwise comparison test sometimes liberalor Dunnett's C pairwise comparison test based on the Studentized range.

Note that these tests are not valid and will not be produced if there are multiple factors in the model. Duncan's multiple range testStudent-Newman-Keuls S-N-Kand Tukey's b are range tests that rank group means and compute a range value. These tests are not used as frequently as the tests previously discussed. The Waller-Duncan t test uses a Bayesian approach.

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