Optimal Scaling of a Data Vector

Description

This package provides tools to perform an optimal scaling analysis on a data vector. The main result of the optimal scaling is a vector of scores which are a least-squares approximation to a vector of quantitative values, subject to measurement constraints based upon a vector of qualitative data values. See Young (1981) for details.

Details

The function that performs the optimal scaling is opscale(). It produces an object of class "opscale". Generic methods are defined for print, summary, and plot (graphing optimally-scaled values versus original data values).

Author(s)

William G. Jacoby

Maintainer: William G. Jacoby <wm.g.jacoby@gmail.com>

References

Young, Forrest W. (1981) “Quantitative Analysis of Qualitative Data.” Psychometrika 46: 357-388.

See Also

opscale,plot.opscale, print.opscale, summary.opscale

Examples

  ###   x1 is vector of qualitative data
  ###   x2 is vector of quantitative values
            x1 <- c(1,1,1,1,2,2,2,3,3,3,3,3,3)     
            x2 <- c(3,2,2,2,1,2,3,4,5,2,6,6,4)     
  ###   Optimal scaling, specifying that x1
  ###   is ordinal-discrete
     op.scaled <- opscale(x.qual=x1, x.quant=x2,   
                  level=2, process=1)              
     print(op.scaled)
     summary(op.scaled)

S3 methods for opscale

Description

Plot, print, shepard, stress, and summary methods for objects of class opscale

Usage

## S3 method for class 'opscale'
plot(x, ...)
## S3 method for class 'opscale'
print(x, ...)
## S3 method for class 'opscale'
summary(object, ...)

Arguments

Details

Method print returns a listing of the data. summary describes the optimal scale transformation. plot calls os.plot and returns an object of class trellis that graphs optimally-scaled values against the original (qualitative) data values.

See Also

os.plot

Examples

  ###   x1 is vector of qualitative data
  ###   x2 is vector of quantitative values
            x1 <- c(1,1,1,1,2,2,2,3,3,3,3,3,3)     
            x2 <- c(3,2,2,2,1,2,3,4,5,2,6,6,4)     
  ###   Optimal scaling, specifying that x1
  ###   is ordinal-discrete
     op.scaled <- opscale(x.qual=x1, x.quant=x2,   
                  level=2, process=1)              
     print(op.scaled)
     summary(op.scaled)
     plot(op.scaled)

Public Opinion During the 1992 U.S. Presidential Election

Description

This data set contains several variables from the Center for Political Studies 1992 National Election Study. Observations with missing values on any of the variables have been deleted.

Usage

data(elec92)

Format

A data frame with 1653 observations on the following 7 variables.

caseid

NES case identification number

bush

Respondents feeling thermometer rating of George H. W. Bush

ideol

Respondents ideological self-placement, seven-point scale ranging from 1=extremely liberal to 7=extremely conservative

econ4yr

Respondents judgment whether national economy has gotten better or worse over preceding four years, five-point scale ranging from 1=much better to 5=much worse

party

Respondents party identification, seven-point scale ranging from 0=strong Democrat to 6=strong Republican

choice

Difference in respondents feeling thermometer ratings of Bush and Clinton

clinton

Respondent”s feeling thermometer rating of Bill Clinton

Source

The full data set from which these observations and variables were extracted is available on the Study Page for the American National Election Studies 1992 Time Series Study, at https://electionstudies.org/data-center/1992-time-series-study/.

References

Jacoby, William G. (1999) “Levels of Measurement and Political Research: An Optimistic View.” American Journal of Political Science 43: 271-301.

Examples

library(optiscale)
data(elec92)
summary(lm(choice ~ party + ideol + econ4yr, data = elec92))

Optimal scaling of a data vector

Description

This function produces an object of class "opscale", containing a vector that is a least-squares approximation to a vector of quantitative values, subject to measurement constraints based upon a vector of qualitative data values.

Usage

opscale(x.qual, x.quant = seq(1:length(x.qual)), level = 1,
   process = 1, na.impute = FALSE,
   na.assign = TRUE, rescale = TRUE)

Arguments

Details

The opscale() function operationalizes a measurement theory proposed by Young (1981) in order to facilitate an analysis strategy called “Alternating Least Squares, Optimal Scaling”. The optimal scaling transformation produces a vector (say, OS) that is a least-squares approximation to x.quant, subject to measurement constraints based upon x.qual. If x.qual is nominal level, then the values in OS are the conditional means of x.quant, within distinct categories of x.qual. If x.qual is ordinal level, then the values in OS are the conditional means of x.quant, adjusted to be weakly monotonic to the values in x.qual, using Kruskals (1964b) monotonic transformation. If x.qual is discrete, then all data objects sharing a common value in x.qual must be assigned the same value in OS. If x.qual is continuous, then data objects sharing a common value in x.qual can fall within a closed interval of values in OS. The continuous-discrete measurement process distinction corresponds to Kruskals (1964a) primary and secondary approaches to ties.

Value

The opscale() function returns an object of class "opscale" containing a list with the following components:

References

Kruskal, Joseph B. (1964a) “Multidimensional Scaling by Optimizing Goodness of Fit to a Nonmetric Hypothesis.” Psychometrika 29: 1-27.

Kruskal, Joseph B. (1964b) “Nonmetric Multidimensional Scaling: A Numerical Method.” Psychometrika 29: 115-129.

Young, Forrest W. (1981) “Quantitative Analysis of Qualitative Data.” Psychometrika 46: 357-388.

See Also

plot.opscale, print.opscale, summary.opscale

Examples

  ###   x1 is vector of qualitative data
  ###   x2 is vector of quantitative values
            x1 <- c(1,1,1,1,2,2,2,3,3,3,3,3,3)
            x2 <- c(3,2,2,2,1,2,3,4,5,2,6,6,4)
  ###   Optimal scaling, specifying that x1
  ###   is ordinal-discrete
     op.scaled <- opscale(x.qual=x1, x.quant=x2,
                  level=2, process=1)
     print(op.scaled)
     summary(op.scaled)

Graph of optimal scaling transformation

Description

Line and point plot showing optimally-scaled values on the vertical axis, original data values (assumed to be qualitative) on the horizontal axis.

Usage

   os.plot(x.qual, os.data, 
      main.title = "Plot of optimal transformation")

Arguments

Value

Object of class trellis.

See Also

plot.opscale,

Examples

  ###   x1 is vector of qualitative data
  ###   x2 is vector of quantitative values
            x1 <- c(1,1,1,1,2,2,2,3,3,3,3,3,3)     
            x2 <- c(3,2,2,2,1,2,3,4,5,2,6,6,4)     
  ###   Optimal scaling, specifying that x1
  ###   is ordinal-discrete
     op.scaled <- opscale(x.qual=x1, x.quant=x2,   
                  level=2, process=1)              
  ###   Plot of optimal scaling transformation
     os.plot(op.scaled$qual, op.scaled$os)         

Shepard diagram for opscale

Description

Graph showing data (assumed quantitative) on vertical axis, optimally-scaled data on horizontal axis.

Usage

shepard(x, ...)

shep.plot(x.quant, os.data, main.title = "Shepard Diagram")

Arguments

Value

shepard() and shep.plot() both produce an object of class trellis

Warning

If using shep.plot(), the Shepard diagram should be created using "raw" optimally scaled values. That is, the OS values should NOT be rescaled to the mean and standard deviation of the original qualitative data.

Examples

  ###   x1 is vector of qualitative data
  ###   x2 is vector of quantitative values
            x1 <- c(1,1,1,1,2,2,2,3,3,3,3,3,3)     
            x2 <- c(3,2,2,2,1,2,3,4,5,2,6,6,4)     
  ###   Optimal scaling, specifying that x1
  ###   is ordinal-discrete, optimally scaled
  ###   values are not rescaled
     op.scaled <- opscale(x.qual=x1, x.quant=x2,   
                  level=2, process=1,
                  rescale=FALSE)              
  ###   Create Shepard diagram
     shepard(op.scaled)
  ###   Same results are produced by:
     shep.plot(op.scaled$quant, op.scaled$os)       

Stress coefficients for opscale

Description

Calculates stress coefficients summarizing lack of fit between two vectors.

Usage

stress(x, ...)

calc.stress(quant, os, rescale = FALSE, 
   os.raw.mean = mean(os, na.rm = TRUE), 
   os.raw.sd = sd(os, na.rm = TRUE))

Arguments

Value

stress() and calc.stress() both produce a vector with three elements:

Warning

If using calc.stress(), the stress coefficients must be created using "raw" optimally scaled values. That is, the OS values should NOT be rescaled to the mean and standard deviation of the original qualitative data.

Examples

  ###   x1 is vector of qualitative data
  ###   x2 is vector of quantitative values
            x1 <- c(1,1,1,1,2,2,2,3,3,3,3,3,3)     
            x2 <- c(3,2,2,2,1,2,3,4,5,2,6,6,4)     
  ###   Optimal scaling, specifying that x1
  ###   is ordinal-discrete, optimally scaled 
  ###   values are not rescaled
     op.scaled <- opscale(x.qual=x1, x.quant=x2,   
                  level=2, process=1,
                  rescale=FALSE)              
  ###  Calculate stress coefficients
    stress(op.scaled)
  ###   Same results can be obtained with:
    calc.stress(op.scaled$quant, op.scaled$os)