Main effects of mother talk at time 1

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Measurement Invariance Why and How?
Rosie Ensor, Claire Hughes, Martha Hart and Anji
Wilson
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Measurement Invariance an important (and easy to
test) pre-requisite for many analyses

• Why?
• Equivalence of measurement characteristics of
  indicators over time are necessary (but not
  sufficient) to demonstrate true change
• E.g. Each decade, IQ scores increase (Flynn
  effect) but scores are not comparable over time
• It is also important to evaluate equivalence of
  measurement characteristics of indicators across
  groups
• E.g. IQ scores sometimes differ according to
  ethnicity, but scores are not necessarily
  comparable between groups
• How?
• Confirmatory Factor Analysis (CFA) parameters
  (unstandardized) can be restricted to be equal in
  value
• Indicators have the same metric if parameters are
  equivalent
• CFAs with equality constraints are nested models
  and so can be evaluated using ?2 difference tests

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How to test longitudinal measurement invariance

• Step 1 model the same factor structure at both
  time-points
• Step 2 constrain like indicator factor loadings
  to be equal
• Use ?2 difference test to evaluate if
  constraints significantly degrade model fit
• If not, a 1 unit increase in the latent construct
  reflects the same increase in repeated indicators
• Step 3 place equality constraints on like
  indicator intercepts
• Use ?2 difference test to evaluate if constraints
  significantly degrade model fit
• If not, at a constant level of latent construct,
  the repeated indicators have a similar score
• Step 4 test the equality of like indicator error
  variances
• ?2 difference test will show a significant
  decrease in model fit
• Not as important to evaluation of measurement
  invariance as prior steps

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Example analysis and syntaxTracking Executive
Function Across the Transition to School Hughes,
C., Ensor, R., Wilson, A. and Graham, A. (under
review)

• At ages 4 and 6, 190 children completed planning,
  inhibitory control and working memory tests
• Good performance on EF tasks requires many
  non-executive processes
• Having adjusted to structured school environment,
  children may cope more readily with peripheral
  test demands

critical value of ?2 (2) 5.99, p .05

• Equal structure ?2 (8) 10.95
• Equal loadings ?2 (10) 14.97, ?2diff (2)
  4.02, ns
• Equal intercepts ?2 (12) 17.90, ?2diff (2)
  2.93, ns

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How to test measurement invariance across two or
more groups

• Alternative terms for across-group measurement
  invariance
• Equal factor structure configural invariance
• Equal factor loadings metric / weak factorial
  invariance
• Equal indicator intercepts scalar / strong
  factorial invariance
• Equal indicator residuals strict factorial
  invariance
• Establishing measurement invariance enables tests
  of population heterogeneity Do structural
  parameters vary across groups?
• CFA with covariates indicators and / or latent
  factors are regressed onto a dummy variable
  denoting group membership
• Significant direct effect of covariate on
  indicators variant intercepts
• Significant effect of covariate on latent factor
  group difference in factor means
• Multiple-groups CFA simultaneous analysis of CFA
  in 2 (or more) groups
• Two separate input matrices and measurement
  models
• Similar procedures to those for testing
  longitudinal measurement invariance
• Also, can constrain factor variances, factor
  covariances, latent means in both groups

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Example CFA with covariatesChildrens Problem
Behaviors with Siblings and Friends Hughes, C.,
Hart, J.M., Wilson, A. and Ensor, R. (under
review)

• Observations of antisocial behaviours within
  structured play are sometimes more ecologically
  valid for boys than girls
• 97 6 year olds (56 boys) aggression, disruption,
  arousal and negative affect were rated using
  4-point scales from videos of interaction during
  marbles and walk the plank games with friends and
  siblings
• Significant direct effect of gender on three
  indicators z 2.37
• At any given value of latent construct of problem
  behaviours while playing marbles game with
  friends
• boys gt girls for aggression (.27 units) and
  disruption (.75 units)
• girls gt boys for arousal (.12 units)

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Example multiple-groups CFAChildrens Problem
Behaviors with Siblings and Friends Hughes, C.,
Hart, J.M., Wilson, A. and Ensor, R. (under
review)

• Observation data may reflect context specificity
  and day-to-day variability
• 4 groups ? Marbles / Walk the Plank with
  Siblings / Friends
• Measurement characteristics of indicators in each
  group compared with those of other 3 groups
  combined
• for example Marbles with Siblings vs. other
  three groups

• If indicator means included in model, MPlus
  default is to hold factor loadings and intercepts
  to equality across groups

• Equal structure ?2 (4) 1.4
• Equal loadings ?2 (7) 3.67, ?2diff (3) 2.27,
  ns
• Equal intercepts ?2 (10) 6.72, ?2diff (3)
  3.05, ns

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Partial measurement invariance

• If the equality constraints on a family of
  parameters (e.g., factor loadings) leads to a
  significant increase in ?2
• Establish whether a particular constrained
  parameter has a high impact
• Modify the model by freely estimating the
  potentially variant parameter
• Test whether the modified model leads to a
  significant increase in ?2
• Further tests of invariance can proceed in
  context of partial measurement invariance
• Minimum requirement other than marker, at least
  one invariant indicator
• Example multiple-groups CFA Marbles with Friends
  vs. other 3 groups
• Equal structure ?2 (4) 2.38
• All like loadings were constrained to equality
  ?2 (7) 13.46, ?2diff (3) 11.08, p lt. 05
• Aggression was marker indicator, negative affect
  was freely estimated, arousal and disruption were
  constrained to equality ?2 (6) 3.77, ?2diff
  (2) 1.39, ns
• Equal intercepts ?2 (9) 9.03, ?2diff (3)
  5.26, ns

critical value of ?2 (2) 5.99, p .05 ?2 (3)
7.81, p .05
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Conclusions

• The examination of measurement invariance should
  precede analyses of longitudinal data
• latent growth curve models
• autoregressive / cross-lagged models
• Comparison of group latent means is analogous to
  ANOVA but preceding tests of measurement
  invariance indicate appropriateness of analysis
  of group differences
• Structural parameters can be analysed, even in
  the context of partial measurement invariance