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Choosing appropriate statistical test for ordinal data

Choosing appropriate statistical test for ordinal data


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Ordinal response variables show up often in biology, but I'm not sure how they are best analysed. Some examples are qualitative assessments (little, many, a lot) or risk assessments (low risk, medium risk, high risk).

My study is about correlation between pain and depression levels. We are measuring pain on an ordinal Likert scale. This is a number scale running from 10 (severe pain) to 0 (no pain).

We will also measure depression on a similar scale (scale: 4 - 0).

What statistical test should I use?


You can use ordinal multinomial regression (also known as ordered logit) if the response is ordered. These methods are basically extensions of logistic regressions, but using e.g. a cumulative logit instead of the logit. However, there are a number of different assumptions you need to consider. For instance, are you going to use a proportional-odds assumption (which is commonly used), which means that there is an equal probability of going from e.g. class 1 -> 2 and class 5 -> 6? You can also evaluate the proportional odds assumption using plots or a score test. If the response cannot be ordered, there are multinomial nominal methods that you can use, and you can also evaluate the results from an ordered analysis by comparing the predictions with ones from a multinomial nominal analysis. I have used these kinds of methods for the analysis is Red list classifications, which are clearly ordered but cannot simply be transformed into a numerical response (similar to your situation).

The book Analysis of Ordinal Categorical Data (Agresti. 2010) is a really good starting point.

In R you can look at the packagespolrandvgamfor ways to perform different types of analyses on ordinal data. The author of the book above has also published some examples for categorical data analysis in R, using the packages I mentioned: Examples of Using R for Modeling Ordinal Data. In SAS similar analyses can be done usingProc GenmodandProc Logistic.


The testing of an ordinal scale requires non-parametric statistical tests. Mean and standard deviation are invalid parameters for descriptive statistics whenever data are on ordinal scales, as are any parametric analyses based on the normal distribution.

The report of Allen & Seaman, 2007 describes a number of possible tests:

Nonparametric procedures-based on the rank, median or range-are appropriate for analyzing these data, as are chi-squared statistics.

Notably, Kruskall-Wallis models can be used to replace a standard parametric analysis of variance, as it is based on the ranks and not the means of the responses. Given these scales are representative of an underlying continuous measure, one recommendation is to analyze them as interval data as a pilot prior to gathering the continuous measure.

However, non-parametric tests are notoriously low in statistical power. There is a way of making an ordinal Likert scale like the ones you use continuous by using a ruler or slider, see the following figure (taken from Allen & Seaman, 2007):

This trick makes it continuous and normal parametric tests can be used, ramping up statistical power substantially. It is a lot more work though to analyze the data. Especially when the subjects indicate their perceived pain/depression by physically putting a mark on a paper ruler, as you have to manually measure the responses. A digital slider can make your life easier. If you are planning to do hundreds of subjects you should carefully think about the possible options.

Good luck!


Statistical Testing: How to select the best test for your data?

T his post is not meant for seasoned statisticians. This is geared towards data scientists and machine learning (ML) learners & practitioners, who like me, do not come from a statistical background.

For a person being from a non-statistical background the most confusing aspect of statistics, are the fundamental statistical tests, and when to use which test?. This post is an attempt to mark out the difference between the most common tests and the relevant key assumptions.


This seems to be a two-sample test with Groups 1 (of size $n_1$) and 2 (of size $n_2$). Your data are scores from 1 to 10 on the question.

Welch t test. If $n_1$ and $n_2$ are large enough (perhaps both above 20), you might be able to get a reliable answer using a Welch 2-sample t-test.

Wilcoxon test. You are almost sure to have lots of ties (repeated scores) even if both sample sizes are relatively small. Thus you will get error messages about ties when trying to do a Wilcoxon rank-sum test, along with an approximated P-value or a statement that a P-value is not available (depending of the software you use).

Permutation test. Perhaps it is best to do a permutation test. Under the null hypothesis that the two groups tend to give the same responses to the question, the argument is that the scores could be permuted between Groups A and B without effect. So if we choose some measure of difference such as the difference $D = ar X_1 - ar X_2$ between the two sample means, we can use either combinatorics or simulation to get the null permutation distribution of $D$, and judge whether your observed value of $D$ is consistent with the null distribution.

Example. I will illustrate each kind of test using fake data with 25 subjects in each group (although none of the tests require sample sizes to be equal).

Here are listings and summaries of some fake data to use for testing.

A quick look shows means to be greater in Group 2 than in Group 1. Is this difference statistically significant?

t test: A Welch 2-sample t test in R statistical software finds a significant difference. (P-value $approx$ 2%.) The only doubt is whether data are sufficiently nearly normally distributed for the t test to give accurate results. (Data for both groups spectacularly fail a Shapiro-Wilk test with P-values < .01. But sample sizes may be large enough for the t test to be useful anyhow.)

Wilcoxon test: The Wilcoxon test, for a difference in medians gives a (tentative) P-value of about 2%, but warns that it may not be accurate. However, there are only seven uniquely different values among the 50 subjects. So the number of ties is 'massive' and the Wilcoxon test is based on a comparison of ranks, which can be problematic when there are many ties. I would not want to trust the result of the Wilcoxon test.

Permutation test. It would be tedious to derive the exact permutation distribution of $D$ for this example. The usual cure is to simulate a large number of permutations and to approximate the P-value from simulation results. Here is a brief program in R to find the approximate P-value (2.1%) of the permutation test. (You may get a slightly different P-value at each run of the program, but not enough different to matter in the interpretation. For this program, subsequent runs all gave values rounding to 2%)

Here is a histogram of the approximate permutation distribution. The solid red line at the left is the observed value of $D$ for the data above. The dotted red line at the right is just as extreme (far from 0) as the observed value of $D.$ The P-value of this 2-sided permutation test is the percentage of values in the permutation distribution outside these red lines, in this case, 2.1%.

Conclusion: The two groups differ significantly. The t test is probably OK, because, for samples this large, the Central Limit Theorem tends to make the sample means very nearly normal even if the data are not normal. For groups as small as ten, I would certainly insist on seeing permutation test results before drawing a conclusion.

You can read more about permutation tests in this paper by Eudey. The two-sample test above is discussed, with additional examples, in Section 4.

Almost certainly, your data will look different than my fake data. Please let me know if you have trouble relating my answer to your specific data.

Note: The fake data above were generated from populations with respective means about 3/5 and 5/6 using the R code below. (So it is appropriate that the tests found a significant difference.) By using the same seed I used, you should get exactly the same data.

Addendum (Your Data from Comment). Your result in the Comment seems OK. Significant at 9.3% < 10% level sometimes optimistically called "suggestive" of significance.

If you honestly expected (before seeing data) Gp2 scores to be higher, then maybe this should be a left-sided test of $H_0: mu_1 ge mu_2$ vs. $H_a: mu_1 < mu_2.$ if so, P-value would be 3.8% < 5% for significance at the 5% level.

Welch t-test gives P-value 0.09024. Repeat of permutation test with m = 10^6 iterations to reduce possibility of simulation error.

Note: If this is for a reviewed paper, you might get criticism (as noted by @Nameless) that the permutation test involves taking sample means of ordinal data. Possible nonparametric, ordinal-oriented alternatives:


Choosing appropriate statistical test for ordinal data - Biology

Test of Homogeneity of Poisson Rates

Poisson C.I. Chi-Square Goodness-of-Fit Test Tests for
Goodness-of-Fit
Kolmogorov Test


Tests for Locationt
Wilcoxon Signed Rank Test

Hodges-Lehmann C.I. for Median Paired Samples Sign Test

Marginal Homogeneity Test Wilcoxon Signed Rank Test

Permutation Test with General Scores

Hodges-Lehmann C.I. for Shift Two Independent Samples: Unstratified Conditional Inference
Fisher's Exact Test


Unconditional Inference
Barnard's Test

Poisson Samples
CI for common Poisson Rate-Ratio Tests for Location
Wilcoxon Mann Whitney Test


Tests for Scale
Siegel-Tukey Test


Omnibus Tests
Kolmogorov-Smirnov Test

Wald-Wolfowitz Runs Test Tests for Location
Wilcoxon Mann Whitney Test

Hodges-Lehmann C.I. for Shift


Tests for Scale
Siegel-Tukey Test


Omnibus Tests
Kolmogorov-Smirnov Test


Tests for Censored Survival Data Wilcoxon-Gehan Test

Logrank Test Two Independent Samples: Stratified Test for Homogeneity of Odds-Ratios

Test for Common Odds-Ratio

C.I. for Common Odds-Ratio Poisson Samples
Test for Homogeneity of Poisson Rate-ratio.

CI for common Poisson Rate-Ratio Wilcoxon Rank Sum Test

Permutation Test with Stratum-Specific Scores

Conditional (Post-hoc) Power Tests for Complete Data
Wilcoxon Rank Sum Test

Permutation Test with Stratum-Specific Scores


Tests for Censored Survival Data Wilcoxon-Gehan Test


A frequency table is a good place to start. You can do the count, and relative frequency for each level. Also, the total count, and number of missing values may be of use.

You can also use a contingency table to compare two variables at once. Can display using a mosaic plot too.

I'm going to argue from an applied perspective that the mean is often the best choice for summarising the central tendency of a Likert item. Specifically, I'm thinking of contexts such as student satisfaction surveys, market research scales, employee opinion surveys, personality test items, and many social science survey items.

In such contexts, consumers of research often want answers to questions like:

  • Which statements have more or less agreement relative to others?
  • Which groups agreed more or less with a given statement?
  • Over time, has agreement gone up or down?

For these purposes, the mean has several benefits:

1. Mean is easy to calculate:

  • It is easy to see the relationship between the raw data and the mean.
  • It is pragmatically easy to calculate. Thus, the mean can be easily embedded into reporting systems.
  • It also facilitates comparability across contexts, and settings.

2. Mean is relatively well understood and intuitive:

  • The mean is often used to report central tendency of Likert items. Thus, consumers of research are more likely to understand the mean (and thus trust it, and act on it).
  • Some researchers prefer the, arguably, even more intuitive option of reporting the percentage of the sample answering 4 or 5. I.e., it has the relatively intuitive interpretation of "percentage agreement". In essence, this is just an alternative form of the mean, with 0, 0, 0, 1, 1 coding.
  • Also, over time, consumers of research build up frames of reference. For example, when you're comparing your teaching performance from year to year, or across subjects, you build up a nuanced sense of what a mean of 3.7, 3.9, or 4.1 indicates.

3. The mean is a single number:

  • A single number is particularly valuable, when you want to make claims like "students were more satisfied with Subject X than Subject Y."
  • I also find, empirically, that a single number is actually the main information of interest in a Likert item. The standard deviation tends to be related to the extent to which the mean is close to the central score (e.g., 3.0). Of course, empirically, this may not apply in your context. For example, I read somewhere that when You Tube ratings had the star system, there were a large number of either the lowest or the highest rating. For this reason, it is important to inspect category frequencies.

4. It doesn't make much difference

  • Although I have not formally tested it, I would hypothesise that for the purpose of comparing central tendency ratings across items, or groups of participants, or over time, any reasonable choice of scaling for generating the mean would yield similar conclusions.

For basic summaries, I agree that reporting frequency tables and some indication about central tendency is fine. For inference, a recent article published in PARE discussed t- vs. MWW-test, Five-Point Likert Items: t test versus Mann-Whitney-Wilcoxon.

For more elaborated treatment, I would recommend reading Agresti's review on ordered categorical variables:

It largely extends beyond usual statistics, like threshold-based model (e.g. proportional odds-ratio), and is worth reading in place of Agresti's CDA book.

Below I show a picture of three different ways of treating a Likert item from top to bottom, the "frequency" (nominal) view, the "numerical" view, and the "probabilistic" view (a Partial Credit Model):

The data comes from the Science data in the ltm package, where the item concerned technology ("New technology does not depend on basic scientific research", with response "strongly disagree" to "strongly agree", on a four-point scale)

Conventional practice is to use the non-parametric statistics rank sum and mean rank to describe ordinal data.

assign a rank to each member in each group

e.g., suppose you are looking at goals for each player on two opposing football teams then rank each member on both teams from first to last

calculate rank sum by adding the ranks per group

the magnitude of the rank sum tells you how close together the ranks are for each group

M/R is a more sophisticated statistic than R/S because it compensates for unequal sizes in the groups you are comparing. Hence, in addition to the steps above, you divide each sum by the number of members in the group.

Once you have these two statistics, you can, for instance, z-test the rank sum to see if the difference between the two groups is statistically significant (I believe that's known as the Wilcoxon rank sum test, which is interchangeable, i.e., functionally equivalent to the Mann-Whitney U test).

R Functions for these statistics (the ones I know about, anyway):

wilcox.test in the standard R installation

meanranks in the cranks Package

Based on the abstract This article may be helpful for comparing several variables that are Likert scale. It compares two types of non-parametric multiple comparison tests: One based on ranks and one based on a test by Chacko. It includes simulations.

I usually like to use Mosaic plot. You can create them by incoorporating other covariates of interest (such as: sex, stratified factors etc.)

I agree with Jeromy Anglim's evaluation. Remember that Likert responses are estimates &mdash you are not using a perfectly reliable ruler to measure a physical object with stable dimensions. The mean is a powerful measure when using reasonable sample sizes.

In business and product R&D, the mean is by far the most common statistic used with Likert scales. When using Likert scales I have usually chosen a measure that ideally fits the research question. For instance, if you are talking about "preference" or "attitudes" you can use multiple Likert-based indicators, with each indicator providing slightly different insight.

To evaluate the question "how will people in segment $i$ react to service offering $X$," I may look at (1) arithmetic mean, (2) exact median, (3) percentage most favorable response (top box), (4) % top two boxes, (5) ratio of top two boxes to bottom two boxes, (6) percentage within mid-range boxes. etc. Each measure tells a different piece of the story. In a very critical project, I use multiple Likert-based indicators. I will also use multiple indicators with small samples and when a specific cross tab has an "interesting" structure or looks information-rich. Ahhh. the art of statistics.


Introduction

Welcome to the third edition of the Handbook of Biological Statistics! This online textbook evolved from a set of notes for my Biological Data Analysis class at the University of Delaware. My main goal in that class is to teach biology students how to choose the appropriate statistical test for a particular experiment, then apply that test and interpret the results. In my class and in this textbook, I spend relatively little time on the mathematical basis of the tests for most biologists, statistics is just a useful tool, like a microscope, and knowing the detailed mathematical basis of a statistical test is as unimportant to most biologists as knowing which kinds of glass were used to make a microscope lens. Biologists in very statistics-intensive fields, such as ecology, epidemiology, and systematics, may find this handbook to be a bit superficial for their needs, just as a biologist using the latest techniques in 4-D, 3-photon confocal microscopy needs to know more about their microscope than someone who's just counting the hairs on a fly's back. But I hope that biologists in many fields will find this to be a useful introduction to statistics.

You may navigate through these pages using the "Previous topic" and "Next topic" links at the top of each page, or you may skip from topic to topic using the links on the left sidebar.

I have provided a spreadsheet to perform many of the statistical tests. Each comes with sample data already entered just download the spreadsheet, replace the sample data with your data, and you'll have your answer. The spreadsheets were written for Excel, but they should also work using the free program Calc, part of the OpenOffice.org suite of programs. If you're using OpenOffice.org, some of the graphs may need re-formatting, and you may need to re-set the number of decimal places for some numbers. Let me know if you have a problem using one of the spreadsheets, and I'll try to fix it.

I've also linked to a web page for each test wherever possible. I found most of these web pages using John Pezzullo's excellent list of Interactive Statistical Calculation Pages, which is a good place to look for information about tests that are not discussed in this handbook.

There are instructions for performing each statistical test in SAS, as well. It's not as easy to use as the spreadsheets or web pages, but if you're going to be doing a lot of advanced statistics, you're going to have to learn SAS or a similar program sooner or later. I've got a page on the basics of SAS.

Salvatore Mangiafico has written An R Companion to the Handbook of Biological Statistics, available as a free set of web pages and also as a free pdf. R is a free statistical programming language, useable on Windows, Mac, or Linux computers, that is becoming increasingly popular among serious users of statistics. If I were starting from scratch, I'd learn R instead of SAS and make my students learn it, too. Dr. Mangiafico's book provides example programs for nearly all of the statistical tests I describe in the Handbook, plus useful notes on getting started in R.

Printed version

While this handbook is primarily designed for online use, you may find it convenient to print out some or all of the pages. If you print a page, the sidebar on the left, the banner, and the decorative pictures (cute critters, etc.) should not print. I'm not sure how well printing will work with various browsers and operating systems, so if the pages don't print properly, please let me know.

If you want a spiral-bound, printed copy of the whole handbook, you can buy one for $18 plus shipping from Lulu.com. I've used this print-on-demand service as a convenience to you, not as a money-making scheme, so please don't feel obligated to buy one.

You can also download a free pdf of the print version. The pdf has page numbers and a table of contents, so it may be a little easier to use than individually printed web pages.

If you use this handbook and want to cite it in a publication, please cite it as:

McDonald, J.H. 2014. Handbook of Biological Statistics, 3rd ed. Sparky House Publishing, Baltimore, Maryland.

It's better to cite the print version, rather than the web pages, so that people of the future can see exactly what you were looking at. If you just cite a web page, it might be quite different by the time someone looks at it a few years from now. If you need to see what someone has cited from an earlier edition, you can download pdfs of the first edition or the second edition.

Pitcher plants, Darlingtonia californica. This is an example of a decorative picture that I hope will brighten your online statistics experience, but you won't waste paper by printing it.

I am constantly trying to improve this textbook. If you find errors, broken links, typos, or have other suggestions for improvement, please e-mail me at [email protected] If you have statistical questions about your research, I'll be glad to try to answer them. However, I must warn you that I'm not an expert in all areas of statistics, so if you're asking about something that goes far beyond what's in this textbook, I may not be able to help you. And please don't ask me for help with your statistics homework (unless you're in my class, of course!).


Thanks!

Acknowledgments

Preparation of this handbook has been supported in part by a grant to the University of Delaware from the Howard Hughes Medical Institute Undergraduate Science Education Program.

Thanks to Naomi Touchet for helping me with some tricky html and css problems (but don't blame her for the clunky mid-1990s design and "artisanal" html coding, that's all my fault).

Reference

Picture of Darlingtonia californica from one of my SmugMug galleries.

Banner photo

The photo in the banner at the top of each page is three Megalorchestia californiana, amphipod crustaceans which live on sandy beaches of the Pacific coast of North America. They are climbing on my slide rule, which I won in a middle-school math contest. This illustration has been heavily photoshopped to see the original, go to my SmugMug page.

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This page was last revised December 4, 2014. Its address is http://www.biostathandbook.com/index.html . It may be cited as:

McDonald, J.H. 2014. Handbook of Biological Statistics (3rd ed.). Sparky House Publishing, Baltimore, Maryland. This web page contains the content of pages 1-2 in the printed version.

©2014 by John H. McDonald. You can probably do what you want with this content see the permissions page for details.


Choosing appropriate statistical test for ordinal data - Biology

Using SPSS for Ordinally Scaled Data:
Mann-Whitney U, Sign Test, and Wilcoxon Tests

This tutorial will show you how to use SPSS version 9.0 to perform Mann Whitney U tests, Sign tests and Wilcoxon matched-pairs signed-rank tests on ordinally scaled data.

  • Downloaded the standard class data set (click on the link and save the data file)
  • Started SPSS (click on Start | Programs | SPSS 9.0 for Windows)
  • Loaded the standard data set
  1. The dependent variable must be as least ordinally scaled.
  2. The independent variable has only two levels.
  3. A between-subjects design is used.
  4. The subjects are not matched across conditions.
  • The dependent variable is ordinally scaled instead of interval or ratio.
  • The assumption of normality has been violated in a t-test (especially if the sample size is small.)
  • The assumption of homogeneity of variance has been violated in a t-test

In this example, we will determine if people who intend to get a Ph.D. or Psy.D. in psychology are more likely to rely on a calendar or day-planner to remember what they are supposed to be doing (i.e., are people who might become professors more absent minded than other people?)

SPSS assumes that the variable that specifies the category is numeric. In the sample data set, the PhD variable corresponds to the question described above, but it is a string variable. So we will have to recode the variable before we can perform the Mann-Whitney U test. If you don't remember how to automatically recode a variable, see the tutorial on transforming variables. I automatically recoded the PhD variable into a variable called PhDNUM.

  1. Write the hypotheses:
    H 0 : µ PhD µ No PhD
    H 1 : µ PhD < µ No PhD
    Note how the question on the questionnaire is worded. People who respond with a low number (1 = strongly agree) rely on a calendar more than people who respond with a higher number. Thus the hypotheses are "reversed" from what one might think.
  2. Determine if the hypotheses are one- or two-tailed. These hypotheses are one-tailed as the null is written with a greater than or equal to sign.
  3. Specify the level: = .05
  4. Perform the Mann Whitney U test.

Select Analyze | Nonparametric Tests | 2 Independent Samples:

The Two-Independent-Samples Tests dialog box appears:

Select the dependent variable of interest from the list at the left by clicking on it, and then move it into the Test Variable List by clicking on the upper arrow button. In this example, I selected the variable PLANNER and moved it into the Test Variable List box:

Select the independent variable of interest from the list at the left by clicking on it, and then move it into the Grouping Variable box by clicking on the lower arrow button. In this example, I selected the variable I automatically recoded earlier (PhDNUM) and moved it into the Grouping Variable box:

Next, we must define the groups of the independent variable. Click on the Define Groups button that is just below the Grouping Variable box. The Two Independent Samples: Define Groups dialog box appears:

Enter the value that corresponds to one level of the independent variable in the Group 1 box and the value that corresponds to the other level of the independent variable in the Group 2 box. Since we automatically recoded the PhD variable into the PhDNUM variable, the people who responded No to the question have a value of 1 and the people who responded Yes to the question have a value of 2 (see the SPSS output of the automatic recode operation.) Thus we should enter 1 for group 1 and 2 for group 2:

Click on the Continue button in the Two Independent Samples: Define Groups dialog box. The Two-Independent Samples Test dialog box should be on top now. Make sure that the Mann-Whitney U option is selected in the Test Type frame. That is, there should be a check mark next in the box to the left of Mann-Whitney U:

Click on the Options button. The Two-Independent-Samples: Options dialog box appears:

Select the Descriptive statistics option by clicking in the box to the left of Descriptives if it does not already have a check mark in it:

Click on the Continue button in the Two-Independent-Samples: Options dialog box. Click on OK in the Two-Independent-Samples Tests box to perform the Mann-Whitney U test. The SPSS output viewer will appear. It should contain three sections:

The first section gives the descriptive statistics for the dependent variable and (less usefully) for the independent variable. In this example, there were 31 people (N) who responded to the PLANNER question. They gave a mean response of 2.42 (between AGREE and UNDECIDED) with a standard deviation of 1.43 (although this number may not be meaningful in this example as standard deviation is not a valid statistic for an ordinally scaled variable.)

The second section of the output shows the number (N) of people in each condition (8 people do not intend to get a Ph.D. or Psy.D in psychology and 23 people do) and the mean rank and sum of ranks for each group (useful if you were calculating the U statistic by hand.)

Sign Test and Wilcoxon Matched-Pairs Signed-Rank Test

Both the sign test and the Wilcoxon matched-pairs signed-rank tests are nonparametric statistic that can be used with ordinally (or above) scaled dependent variable when the independent variable has two levels and the participants have been matched or the samples are correlated. Thus, both are useful when a t-test cannot be employed because its assumptions have been violated.

The sign test uses only directional information while the Wilcoxon test uses both direction and magnitude information. Thus the Wilcoxon test is more powerful statistically than the sign test. However, the Wilcoxon test assumes that the difference between pairs of scores is ordinally scaled, and this assumption is difficult to test.

In this example, we will use a fictional data set that is available from here. In this data set, people were matched on their GPA prior to being assigned to one of two conditions: either they were allowed to use an on-line quiz program or they were not allowed to use it. At the end of the semester, the students rated how much they liked the class on a 7-point Likert scale with 1 being that they did not like the class at all and 7 being that they liked the class very much. Notice how the data have been entered into SPSS. There are two variables -- one for the liking score for the people who had the on-line quiz and one for the liking score for the people who did not have the on-line quiz. The data points in each row are matched. That is the two people who gave us scores in the first row have similar GPAs. The two people who gave us scores in the second row have similar GPAs and so on.

We will determine if the mean liking rating is different for the two groups of students.

  1. Write the hypotheses:
    H 0 : µ Quiz = µ No Quiz
    H 1 : µ Quiz µ No Quiz
  2. Determine if the hypotheses are one- or two-tailed. These hypotheses are two-tailed as the null is written with an equal sign.
  3. Specify the level: = .05
  4. Perform the sign test and / or Wilcoxon matched-pairs signed-rank test.

Select Analyze | Nonparametric Tests | 2 Related Samples:

The Two-Related-Samples Tests dialog box appears:

Select the dependent variable that corresponds to one of the means in the hypothesis from the list at the left by clicking on it. Select the other dependent variable that corresponds to the other mean in the hypothesis from the list at the left by clicking on it as well. You should have two variables highlighted. In this example, the first variable (Liking Rating for On-Line Quiz) corresponds to the first mean in the hypothesis, so I clicked on it. The second variable (Liking Rating for No On-Line Quiz) corresponds to the second mean in the hypothesis, so I clicked on it as well:

Move the selected pair of variables into the Test Pair(s) List box by clicking on the arrow button:

Select the type of statistical test that you want to perform in the Test Type section of the dialog box. I will select to perform both the Sign test and the Wilcoxon test:

Click on the Options button. The Two-Related-Samples: Options dialog box appears:

Select the Descriptive statistics option by clicking in the box to the left of Descriptives if it does not already have a check mark in it:

Click on the Continue button in the Two-Related-Samples: Options dialog box. Click on OK in the Two-Related-Samples Tests box to perform the Sign and Wilcoxon tests. The SPSS output viewer will appear. It should contain five sections:

The first section gives the descriptive statistics for the dependent variable for each level of the independent variable. In this example, there were 12 people (N) in each condition. The On-Line quiz people gave a mean liking rating of 6.000 with a standard deviation of 1.0445 (although this number may not be meaningful in this example as standard deviation is not a valid statistic for an ordinally scaled variable.) The No On-Line quiz people gave a mean liking rating of 4.500 with a standard deviation of 1.3143.

The second section of the output shows the ranks for the Wilcoxon test. It gives the number of observations (N), 8, in which the No On-Line Quiz people liked the class less than their matched counterpart (The Negative Ranks row). It also gives the number of observations, 0, in which the No On-Line Quiz people liked the class more than their matched counterparts (the Positive Ranks row.) Finally, it gives the number of observations, 4, in which the No On-Line Quiz people liked the class the same amount as their matched counterparts in the On-Line Quiz group (the Ties row.)

The third section of the output gives the values of the Wilcoxon test. The p value associated with the Wilcoxon test is given at the intersection of the row labeled Asymp. Sig. (2-tailed) (asymptotic significance, 2-tailed) and the column labeled with the difference of the variables that correspond to the means in the hypothesis (e.g. Liking Rating for No On-Line Quiz - Liking Rating for On-Line Quiz. In this example, the p value for the Wilcoxon test is .011.

This section of the output is similar to the ranks section. It is produced for the sign test, while the ranks section is produced for the Wilcoxon test. It gives the number of observations (N), 8, in which the No On-Line Quiz people liked the class less than their matched counterpart (the Negative Differences row). It also gives the number of observations, 0, in which the No On-Line Quiz people liked the class more than their matched counterparts (the Positive Differences row). Finally, it gives the number of observations, 4, in which the No On-Line Quiz people liked the class the same amount as their matched counterparts in the On-Line Quiz group (the Ties row.)


AQA AS/A2 Biology 7401/7402 detailed statistics overview sheet

I am a teacher of Biology and Chemistry based in Devon. Currently teaching in Exeter. Currently teaching AQA iGCSE to Year 11 pupils, just switched to Edexcel iGCSE for Year 9 & 10 pupils and teaching AQA Biology (new specification) to AS/A2 pupils. I try to make engaging resources that involve a mix of collaborative activities as well as resources to encourage pupils to work independently. Also interested in promoting more of an awareness of mental health. Twitter: @missjmbooth

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Empty reply does not make any sense for the end user

Empty reply does not make any sense for the end user

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I am a teacher of Biology and Chemistry based in Devon. Currently teaching in Exeter. Currently teaching AQA iGCSE to Year 11 pupils, just switched to Edexcel iGCSE for Year 9 & 10 pupils and teaching AQA Biology (new specification) to AS/A2 pupils. I try to make engaging resources that involve a mix of collaborative activities as well as resources to encourage pupils to work independently. Also interested in promoting more of an awareness of mental health. Twitter: @missjmbooth


What is interval data?

As we discussed earlier, interval data are a numerical data type. In other words, it’s a level of measurement that involves data that’s naturally quantitative (is usually measured in numbers). Specifically, interval data has an order (like ordinal data), plus the spaces between measurement points are equal (unlike ordinal data).

Sounds a bit fluffy and conceptual? Let’s take a look at some examples of interval data:

  • Credit scores (300 – 850)
  • GMAT scores (200 – 800)
  • IQ scores
  • The temperature in Fahrenheit

Importantly, in all of these examples of interval data, the data points are numerical, but the zero point is arbitrary. For example, a temperature of zero degrees Fahrenheit doesn’t mean that there is no temperature (or no heat at all) – it just means the temperature is 10 degrees less the 10. Similarly, you cannot achieve a zero credit score or GMAT score.

In other words, interval data is a level of measurement that’s numerical (and you can measure the distance between points), but that doesn’t have a meaningful zero point – the zero is arbitrary.

Long story short – interval-type data offers a more sophisticated level of measurement than nominal and ordinal data, but it’s still not perfect. Enter, ratio data…


Conclusion

When dealing with statistical data, it is important to know whether the data you are dealing with is nominal or ordinal, as this information helps you decide how to use the data. A statistician is able to make a proper decision on what statistical analysis to apply to a given data set based on whether it is nominal or ordinal.

The first step to proper identification of nominal and ordinal data is to know their respective definitions. After which, you need to identify their similarities and differences so as not to mix them up during analysis.

This knowledge is very essential, as it helps a researcher determine the type of data that needs to be collected.


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