Is partial measurement invariance testing with 2 items possible?

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Sjoukje Goldman

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Feb 22, 2019, 7:13:34 AM2/22/19
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In my online questionnaire I have three questions to measure a construct called risk. When testing the partial measurement invariance, 2 of the 3 items turn out to have a significant difference across the countries (the questionnaire was completed by respondents from 3 different countries). The output also contains several warning messages which I find difficult to interpret. 

When I drop 1 item and want to test again I get the error model did not converge. Is it possible to test measurement invariance with 2 items? 

The construct risk is an important variable in my conceptual model and I would like to be able to include it. Does anyone have suggestions or tips?

> conf <- "
+ Risk =~ NA*Q7_1 + Q7_2 + Q7_3
+ "
> weak <- "
+ Risk =~ NA*Q7_1 + Q7_2 + Q7_3
+ "
> configural <- cfa(conf, data = DSCP_new, std.lv = TRUE, group="Country")
> weak <- cfa(weak, data = DSCP_new, group="Country", group.equal="loadings")
Warning message:
In lav_model_vcov(lavmodel = lavmodel, lavsamplestats = lavsamplestats,  :
  lavaan WARNING:
    The variance-covariance matrix of the estimated parameters (vcov)
    does not appear to be positive definite! The smallest eigenvalue
    (= 3.516467e-19) is close to zero. This may be a symptom that the
    model is not identified.
> models <- list(fit.configural = configural, fit.loadings = weak)
> partialInvariance(models, "metric") #Q7_1 and Q7_2 significant

$`estimates`
            poolest    load:3   load:2   load:1     std:3     std:2     std:1 diff_std:2 vs. 3 diff_std:1 vs. 3
Risk=~Q7_1 1.046122 0.9549009 1.042294 1.107101 0.7576693 0.8270119 0.8784330       0.06934259      0.120763732
Risk=~Q7_2 1.053865 1.1812193 1.019875 1.026996 0.9812245 0.8471975 0.8531130      -0.13402701     -0.128111496
Risk=~Q7_3 1.021755 0.9941885 1.067469 0.999383 0.8430777 0.9052201 0.8474827       0.06214241      0.004405007
            q:2 vs. 3   q:1 vs. 3 diff_mean:2 vs. 3 diff_mean:1 vs. 3 low_fscore:2 vs. 3 low_fscore:1 vs. 3
Risk=~Q7_1  0.1878868  0.37814360       0.057444123        0.09535405        -0.09440078         -0.1690917
Risk=~Q7_2 -1.0833202 -1.06198017      -0.003531735       -0.12056076         0.28036640          0.1508070
Risk=~Q7_3  0.2686772  0.01542999      -0.205186663       -0.35193410        -0.33048359         -0.3608159
           high_fscore:2 vs. 3 high_fscore:1 vs. 3
Risk=~Q7_1          0.20928903           0.3597998
Risk=~Q7_2         -0.28742987          -0.3919286
Risk=~Q7_3         -0.07988974          -0.3430523

$results
            free.chi free.df      free.p     free.cfi   fix.chi fix.df        fix.p       fix.cfi  wald.chi wald.df
Risk=~Q7_1  9.764141       2 0.007581301 -0.001770460 20.774822      2 3.081802e-05 -4.281230e-03 21.864248       2
Risk=~Q7_2 12.616815       2 0.001820931 -0.002420957  1.493092      2 4.740009e-01  9.992007e-16  1.491717       2
Risk=~Q7_3  3.896155       2 0.142547844 -0.000432381 17.078190      2 1.956672e-04 -3.438286e-03 17.259112       2
                 wald.p
Risk=~Q7_1 0.0000178747
Risk=~Q7_2 0.4743268323
Risk=~Q7_3 0.0001787440

Warning messages:
1: In lav_model_vcov(lavmodel = lavmodel, lavsamplestats = lavsamplestats,  :
  lavaan WARNING:
    The variance-covariance matrix of the estimated parameters (vcov)
    does not appear to be positive definite! The smallest eigenvalue
    (= 6.419751e-19) is close to zero. This may be a symptom that the
    model is not identified.
2: In lav_model_vcov(lavmodel = lavmodel, lavsamplestats = lavsamplestats,  :
  lavaan WARNING:
    The variance-covariance matrix of the estimated parameters (vcov)
    does not appear to be positive definite! The smallest eigenvalue
    (= 8.220385e-19) is close to zero. This may be a symptom that the
    model is not identified.
3: In lav_model_vcov(lavmodel = lavmodel, lavsamplestats = lavsamplestats,  :
  lavaan WARNING:
    The variance-covariance matrix of the estimated parameters (vcov)
    does not appear to be positive definite! The smallest eigenvalue
    (= 4.963963e-19) is close to zero. This may be a symptom that the
    model is not identified.

Terrence Jorgensen

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Feb 28, 2019, 9:57:37 PM2/28/19
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Is it possible to test measurement invariance with 2 items? 

No.  Invariance of only 1 item is equivalent to the configural model.  Equivalence of 2 items can only be assumed, not tested.


Does anyone have suggestions or tips?

Use more items to measure the construct.

Terrence D. Jorgensen
Assistant Professor, Methods and Statistics
Research Institute for Child Development and Education, the University of Amsterdam

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