cusp-package: Cusp Catastrophe Modeling

In cusp: Cusp-Catastrophe Model Fitting Using Maximum Likelihood

Description

Fits cusp catastrophe to data using Cobb's maximum likelihood method with a different algorithm. The package contains utility functions for plotting, and for comparing the model to linear regression and logistic curve models. The package allows for multivariate response subspace modelling in the sense of the GEMCAT software of Oliva et al.

Details

Package:	cusp
Type:	Package
Version:	2.0
Date:	2008-02-14
License:	GNU GPL v2 (or higher)

This package helps fitting Cusp catastrophy models to data, as advanced in Cobb et al. (1985). The main functions are

cusp	Fit Cobb's Cusp catastrophe model; see example below.
summary.cusp	Summary statistics of cusp model fit.
confint.cusp	Confidence intervals for parameter estimates
plot.cusp	Diagnostic plots for cusp model fit
cusp3d	3D graphical display of cusp model fit (experimental).
dcusp	Density of Cobb's cusp distribtution
pcusp	Cumulative probability function of Cobb's cusp distribution
qcusp	Quantile function of Cobb's cusp distribution
rcusp	Sample from Cobb's cusp distribution.
cusp.logist	Fit logistic model for bifurctation testing (experimental)

Author(s)

Raoul Grasman <[e-mail]>

References

L. Cobb and S. Zacks (1985) Applications of Catastrophe Theory for Statistical Modeling in the Biosciences (article), Journal of the American Statistical Association, 392:793–802.

P. Hartelman (1996). Stochastic Catastrophy Theory. Unpublished PhD-thesis.

H. L. J. van der Maas, R. Kolstein, J van der Pligt (2003). Sudden Transitions in Attitudes, Sociological Methods \& Research, 32:125-152.

Oliva, DeSarbo, Day, \& Jedidi. (1987) GEMCAT : A General Multivariate Methodology for Estimating Catastrophe Models, Behavioral Science, 32:121-137.

R. P. P. P. Grasman, H. L. J. van der Maas, \& E-J. Wagenmakers (2009). Fitting the Cusp Catastrophe in R: A cusp Package Primer. Journal of Statistical Software 32(8), 1-28. URL http://www.jstatsoft.org/v32/i08/.

Examples

set.seed(123)
# fitting cusp to cusp data
x <- rcusp(100, alpha=0, beta=1)
fit <- cusp(y ~ x, alpha ~ 1, beta ~ 1)
print(fit)

# example with regressors
## Not run: 
x1 = runif(150)
x2 = runif(150)
z = Vectorize(rcusp)(1, 4*x1-2, 4*x2-1)
data <- data.frame(x1, x2, z)
fit <- cusp(y ~ z, alpha ~ x1+x2, beta ~ x1+x2, data)
print(fit)
summary(fit)
plot(fit)
cusp3d(fit)

## End(Not run)

# use of OK
npar <- length(fit$par)
## Not run: 
while(!fit$OK) # refit if necessary until convergence is OK
    fit <- cusp(y ~ z, alpha ~ x1+x2, beta ~ x1+x2, data, start=rnorm(npar))

## End(Not run)

## Not run: 
# example 1 from paper
data(attitudes)
data(attitudeStartingValues)
fit.attitudes <- cusp(y ~ Attitude, alpha ~ Orient + Involv, beta ~ Involv,
 data = attitudes, start=attitudeStartingValues)

summary(fit.attitudes)
plot(fit.attitudes)
cusp3d(fit.attitudes, B = 0.75, Y = 1.35, theta = 170, phi = 30, Yfloor = -9)

## End(Not run)

Example output

Call:  cusp(formula = y ~ x, alpha = alpha ~ 1, beta = beta ~ 1) 

Coefficients:
a[(Intercept)]  b[(Intercept)]  w[(Intercept)]            w[x]  
       -0.1385          1.4992         -0.1348          1.0584  

Degrees of Freedom: 99 Total (i.e. Null);  96 Residual
Null Deviance:	    129.4 
Delay Deviance:	 34.52 	AIC: 261.5 

Call:  cusp(formula = y ~ z, alpha = alpha ~ x1 + x2, beta = beta ~      x1 + x2, data = data) 

Coefficients:
a[(Intercept)]           a[x1]           a[x2]  b[(Intercept)]           b[x1]  
      -1.84895         2.96779         0.45627        -0.59675         1.00533  
         b[x2]  w[(Intercept)]            w[z]  
       3.76966         0.04711         1.09503  

Degrees of Freedom: 149 Total (i.e. Null);  142 Residual
Null Deviance:	    307.1 
Delay Deviance:	 77.1 	AIC: 223.1 

Call:
cusp(formula = y ~ z, alpha = alpha ~ x1 + x2, beta = beta ~ 
    x1 + x2, data = data)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-2.27600  -0.28075   0.04203   0.33374   2.41136  

Coefficients:
               Estimate Std. Error z value Pr(>|z|)    
a[(Intercept)] -1.84895    0.35112  -5.266 1.40e-07 ***
a[x1]           2.96779    0.46664   6.360 2.02e-10 ***
a[x2]           0.45627    0.29906   1.526    0.127    
b[(Intercept)] -0.59675    0.57007  -1.047    0.295    
b[x1]           1.00533    0.66945   1.502    0.133    
b[x2]           3.76966    0.50438   7.474 7.79e-14 ***
w[(Intercept)]  0.04711    0.06231   0.756    0.450    
w[z]            1.09503    0.04144  26.426  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


  Null deviance: 307.100  on 149  degrees of freedom
Linear deviance: 136.567  on 146  degrees of freedom
Logist deviance:      NA  on  NA  degrees of freedom
 Delay deviance:  77.104  on 142  degrees of freedom

             R.Squared    logLik npar      AIC     AICc      BIC
Linear model 0.4667668 -205.8043    4 419.6086 419.8845 431.6512
Cusp model   0.7519607 -103.5558    8 223.1116 224.1329 247.1967
---
Note: R.Squared for cusp model is Cobb's pseudo-R^2. This value
      can become negative.

	Chi-square test of linear vs. cusp model

X-squared = 204.5, df = 4, p-value = 0

Number of optimization iterations: 46


Call:
cusp(formula = y ~ Attitude, alpha = alpha ~ Orient + Involv, 
    beta = beta ~ Involv, data = attitudes, start = attitudeStartingValues)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-2.67547  -0.71609  -0.03732   0.55902   2.62736  

Coefficients:
               Estimate Std. Error z value Pr(>|z|)    
a[(Intercept)]  0.15271    0.56218   0.272   0.7859    
a[Orient]       0.46210    0.06414   7.204 5.83e-13 ***
a[Involv]       0.09736    0.05445   1.788   0.0738 .  
b[(Intercept)]  0.12397    0.38643   0.321   0.7484    
b[Involv]       0.22738    0.02993   7.597 3.02e-14 ***
w[(Intercept)]  0.10543    0.24485   0.431   0.6668    
w[Attitude]     0.87758    0.06682  13.134  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


  Null deviance: 1067.4  on 1386  degrees of freedom
Linear deviance: 1204.6  on 1383  degrees of freedom
Logist deviance:     NA  on   NA  degrees of freedom
 Delay deviance: 1183.2  on 1380  degrees of freedom

               R.Squared    logLik npar      AIC     AICc      BIC
Linear model  0.13086701 -1870.298    4 3748.596 3748.625 3769.535
Cusp model   -0.09743411 -1804.883    7 3623.765 3623.847 3660.410
---
Note: R.Squared for cusp model is Cobb's pseudo-R^2. This value
      can become negative.

	Chi-square test of linear vs. cusp model

X-squared = 130.8, df = 3, p-value = 0

Number of optimization iterations: 21

cusp documentation built on May 2, 2019, 6:51 p.m.