R Dataset / Package psych / Thurstone.33

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Description

Describes how to create a bar plot based on count data. For an example of count data, see the email50 curated data set which was taken from the Open Intro AHSS textbook (not affiliated). An example of count data in this dataset would be the spam column.

Usage

Select one (1) column to create its barplot and then click 'Submit'. If you do not choose count data, you may get unexpected results.

Students may also be interested in creating barplots for contingency tables.

For a stacked side-by-side barplot, see the other barplot app.

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Usage

Select 1 (one) column from a contingency table like the Gender and Politics or VADeaths curated datasets.

If you do not choose a contingency table, you may get unexpected results. You can import a dataset if you are logged-in.

Details

Shows the student how to create a single stacked bar plot based on a column in a contingency table.

For a basic barplot (single column) based on count data see the count data barplot app.

For a stacked side-by-side barplot see the other stacked barplot app for categorical data.

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Usage

Select 1 (one) column from a contingency table. If you don't have your own dataset, you can choose the Gender and Politics or VADeaths curated datasets. If a contingency table is not chosen, you may get unexpected results.

A contingency table has columns like a regular dataset, but the first row contains row names that categorize and "split-up" the dataset. An example of a contingency table would be something like this:

LIBERAL CONSERVATIVE
F 762 468
M 484 477


This contingency table is take from the Gender and Politics dataset. You can get a preview by selecting the dataset from the Curated Data dropdown above.

Details

This app shows the student how to create a pie chart from a contingency table by hand using a Quadstat dataset.

A pie chart shows proportions of a sample or population. Each piece of a pie chart corresponds to some subset of the sample or population. In this case, we will use the contingency table rows to subset the sample.

Students may also want to view the app for creating a pie chart from count data.

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Usage

Click "Submit" after selecting one column to see how to compute the arithmetic mean (average) of data (vectors).

Description

If all the values of a sample were plotted on a number line, the average would be the point in the middle that would balance the two sides.

The average is greatly influenced by outliers, meaning extreme points can pull the average to the left or right.

If we are referring to the average of population (all observations), the symbol for the average (arithmetic mean) is $\mu$.

If we are referring to the average of a sample (a subset of the population), the symbol for the average (arithmetic mean) is $\bar{x}$.

Computing the average

Suppose we have a sample consisting of $x_1, x_2, x_3,...,x_n$. This means we have $n$ observations. Then,

$$\bar{x}=\frac{x_1, x_2, x_3,...,x_n}{n}.$$

The formula tells us that we need to add all the observations and then divide by the number of observations to compute the mean.

Example 1

Compute the mean of $A = \{1,2,3\}$.

$$\bar{x} = \frac{1+2+3}{3} = 2.$$
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Usage

Select two columns which are to be used in the scatterplot. The first column clicked will be the independent variable (X-axis).

Description

This web application describes how to create a scatterplot of two dataset variables plotted on the xy-axes.

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Median Value

Description

Compute the sample median.

Usage

median(x, na.rm = FALSE, ...)


Arguments

 x an object for which a method has been defined, or a numeric vector containing the values whose median is to be computed. na.rm a logical value indicating whether NA values should be stripped before the computation proceeds. ... potentially further arguments for methods; not used in the default method.

Value

The default method returns a length-one object of the same type as x, except when x is logical or integer of even length, when the result will be double.

If there are no values or if na.rm = FALSE and there are NA values the result is NA of the same type as x (or more generally the result of x[FALSE][NA]).

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

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Attachment Size
836 bytes
Documentation

Seven data sets showing a bifactor solution.

Description

Holzinger-Swineford (1937) introduced the bifactor model of a general factor and uncorrelated group factors. The Holzinger data sets are original 14 * 14 matrix from their paper as well as a 9 *9 matrix used as an example by Joreskog. The Thurstone correlation matrix is a 9 * 9 matrix of correlations of ability items. The Reise data set is 16 * 16 correlation matrix of mental health items. The Bechtholdt data sets are both 17 x 17 correlation matrices of ability tests.

Usage

data(Thurstone)
data(Thurstone.33)
data(Holzinger)
data(Holzinger.9)
data(Bechtoldt)
data(Bechtoldt.1)
data(Bechtoldt.2)
data(Reise)


Details

Holzinger and Swineford (1937) introduced the bifactor model (one general factor and several group factors) for mental abilities. This is a nice demonstration data set of a hierarchical factor structure that can be analyzed using the omega function or using sem. The bifactor model is typically used in measures of cognitive ability.

There are several ways to analyze such data. One is to use the omega function to do a hierarchical factoring using the Schmid-Leiman transformation. This can then be done as an exploratory and then as a confirmatory model using omegaSem. Another way is to do a regular factor analysis and use either a bifactor or biquartimin rotation. These latter two functions implement the Jennrich and Bentler (2011) bifactor and biquartimin transformations. The bifactor rotation suffers from the problem of local minima (Mansolf and Reise, 2016) and thus a mixture of exploratory and confirmatory analysis might be preferred.

The 14 variables are ordered to reflect 3 spatial tests, 3 mental speed tests, 4 motor speed tests, and 4 verbal tests. The sample size is 355.

Another data set from Holzinger (Holzinger.9) represents 9 cognitive abilities (Holzinger, 1939) and is used as an example by Karl Joreskog (2003) for factor analysis by the MINRES algorithm and also appears in the LISREL manual as example NPV.KM.

Another classic data set is the 9 variable Thurstone problem which is discussed in detail by R. P. McDonald (1985, 1999) and and is used as example in the sem package as well as in the PROC CALIS manual for SAS. These nine tests were grouped by Thurstone and Thurstone, 1941 (based on other data) into three factors: Verbal Comprehension, Word Fluency, and Reasoning. The original data came from Thurstone and Thurstone (1941) but were reanalyzed by Bechthold (1961) who broke the data set into two. McDonald, in turn, selected these nine variables from the larger set of 17 found in Bechtoldt.2. The sample size is 213.

Another set of 9 cognitive variables attributed to Thurstone (1933) is the data set of 4,175 students reported by Professor Brigham of Princeton to the College Entrance Examination Board. This set does not show a clear bifactor solution but is included as a demonstration of the differences between a maximimum likelihood factor analysis solution versus a principal axis factor solution.

More recent applications of the bifactor model are to the measurement of psychological status. The Reise data set is a correlation matrix based upon >35,000 observations to the Consumer Assessment of Health Care Provideers and Systems survey instrument. Reise, Morizot, and Hays (2007) describe a bifactor solution based upon 1,000 cases.

The five factors from Reise et al. reflect Getting care quickly (1-3), Doctor communicates well (4-7), Courteous and helpful staff (8,9), Getting needed care (10-13), and Health plan customer service (14-16).

The two Bechtoldt data sets are two samples from Thurstone and Thurstone (1941). They include 17 variables, 9 of which were used by McDonald to form the Thurstone data set. The sample sizes are 212 and 213 respectively. The six proposed factors reflect memory, verbal, words, space, number and reasoning with three markers for all expect the rote memory factor. 9 variables from this set appear in the Thurstone data set.

Two more data sets with similar structures are found in the Harman data set.

• Bechtoldt.1: 17 x 17 correlation matrix of ability tests, N = 212.

• Bechtoldt.2: 17 x 17 correlation matrix of ability tests, N = 213.

• Holzinger: 14 x 14 correlation matrix of ability tests, N = 355

• Holzinger.9: 9 x 9 correlation matrix of ability tests, N = 145

• Reise: 16 x 16 correlation matrix of health satisfaction items. N = 35,000

• Thurstone: 9 x 9 correlation matrix of ability tests, N = 213

• Thurstone.33: Another 9 x 9 correlation matrix of ability items, N=4175

Source

Holzinger: Holzinger and Swineford (1937)
Reise: Steve Reise (personal communication)
sem help page (for Thurstone)

References

Bechtoldt, Harold, (1961). An empirical study of the factor analysis stability hypothesis. Psychometrika, 26, 405-432.

Holzinger, Karl and Swineford, Frances (1937) The Bi-factor method. Psychometrika, 2, 41-54

Holzinger, K., & Swineford, F. (1939). A study in factor analysis: The stability of a bifactor solution. Supplementary Educational Monograph, no. 48. Chicago: University of Chicago Press.

McDonald, Roderick P. (1999) Test theory: A unified treatment. L. Erlbaum Associates. Mahwah, N.J.

Mansolf, Maxwell and Reise, Steven P. (2016) Exploratory Bifactor Analysis: The Schmid-Leiman Orthogonalization and Jennrich-Bentler Analytic Rotations, Multivariate Behavioral Research, 51:5, 698-717, DOI: 10.1080/00273171.2016.1215898

Reise, Steven and Morizot, Julien and Hays, Ron (2007) The role of the bifactor model in resolving dimensionality issues in health outcomes measures. Quality of Life Research. 16, 19-31.

Thurstone, Louis Leon (1933) The theory of multiple factors. Edwards Brothers, Inc. Ann Arbor

Thurstone, Louis Leon and Thurstone, Thelma (Gwinn). (1941) Factorial studies of intelligence. The University of Chicago Press. Chicago, Il.

Examples

if(!require(GPArotation)) {message("I am sorry, to run omega requires GPArotation")
} else {
#holz <- omega(Holzinger,4, title = "14 ability tests from Holzinger-Swineford")
#bf <- omega(Reise,5,title="16 health items from Reise")
#omega(Reise,5,labels=colnames(Reise),title="16 health items from Reise")
thur.om <- omega(Thurstone,title="9 variables from Thurstone") #compare with
thur.bf   <- fa(Thurstone,3,rotate="biquartimin")
factor.congruence(thur.om,thur.bf)
}

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Dataset imported from https://www.r-project.org.