It means that for the same score on the latent variable, people in the two groups do not have different intercepts on the observed variables.
Say you're comparing two racial/ethnic groups on a measure of ability that's used in job selection. You find that you don't have scalar invariance for one item. That means that one group finds one question easier than the other group. That means that if you take the total score, you're going to get a biased score. There's an example of that here: https://www.talentqgroup.com/media/84831/policy_assessment_and_the_law-march-2013-.pdf (look at the British Rail example).
Second example: You want to measure depression, so you ask about crying. Women cry more than men, whether they're depressed or not. Women are therefore going to get higher scores on the measure of depression, even if they're equally depressed.
As Maarten points out, your problem is that you have not set the scale of the second model. True, you have more observed variances/covariances than what you need to identify your model, but you still need to provide a point of reference from which other model parameters can be calculated (Brown, 2015).
You can set the scale using one of three methods:
- Marker variable: one factor loading per latent variable is fixed to 1
- Fixed factor: each latent variable's variance is fixed to 1
- Effects-coding: factor loadings for each latent variable are constrained to average 1
Code for each approach (using the lavaan package's HolzingerSwineford1939 dataset) is presented below. The latent variable I've created is nonsensical/poor-fitting, but it has the same number of indicators as your model, so the example will hopefully be more transferable to your situation.
library(lavaan)
#marker-variable; first factor loading fixed to 1 by default
marker.variable<-'f1=~ x1+x2+x3+x4+x5+x6'
summary(output.marker<-cfa(marker.variable, data=HolzingerSwineford1939), fit.measures=TRUE)
#fixed-factor method; manually free first factor loading/fix latent variance to 1
fixed.factor<-'f1=~ NA*x1+x2+x3+x4+x5+x6
f1~~1*f1'
summary(output.fixed<-cfa(fixed.factor, data=HolzingerSwineford1939), fit.measures=TRUE)
#effects coding; manually free first loading/constrain loadings to average 1
effects.coding<-'f1=~ NA*x1+a*x1+b*x2+c*x3+d*x4+e*x5+f*x6
a+b+c+d+e+f==6'
summary(output.effects<-cfa(effects.coding, data=HolzingerSwineford1939), fit.measures=TRUE)
Note that model fit is identical, regardless of which method of scale-setting that you use; the fit in all three models is $\chi^2 (df = 9) = 103.23, ~p < .001$.
Which method you should use largely depends on the nature of your data and your research goals. The marker variable method is a highly arbitrary method of scale-setting. Like Maarten stated, your latent variables will take on the units of their respective marker variables, so this approach is only informative to the extent that your marker variables are especially meaningful, or perhaps represent some "gold standard" indicator of your latent construct.
The fixed factor method, alternatively, is easy to specify, and essentially standardizes your latent variables (if you're examining mean structures, you would fix the latent means to zero as well). Since we standardize variables all the time, this is a highly intuitive and widely acceptable form of scale-setting for latent variables, though the resultant scaling is not inherently meaningful. Even so, it's probably the best method to "default" to, unless you have a strong imperative to use one of the other methods.
Effects-coding is a relative new-comer to methods of scale-setting (see Little, Slegers, & Card, 2006, for a thorough discussion). It's greatest advantage is when you are modeling latent means. When doing so, you would also constrain item intercepts to average 0. The effect of these constraints is that your latent variables will be on the exact same scale as your original items. For example, if the average of your indicators was "5", your latent mean would also be "5", though your latent variance would be smaller than you observed variance. Because the constraints on the loadings and intercepts can be more computationally demanding, especially in more complicated models, and occasionally result in convergence errors, effects-coding is probably not worth it unless you plan to examine latent means. But for the particular purpose of examining latent means, it's great.
References
Brown, T. A. (2015). Confirmatory factor analysis for applied research (2nd Edition). New York, NY: Guilford Press.
Little, T. D., Slegers, D. W., & Card, N. A. (2006). A non-arbitary method of identifying and scaling latent variables in SEM and MACS models. Structural Equation Modeling, 13, 59-72.
Best Answer
The intercept or mean of a latent variable is arbitrary, like the variance, and is usually fixed to zero if you have a single group model (or a single time point model). The intercept of the measured variable is the expected value when the predictor (the latent variable) is equal to zero.
You anchor the mean of the latent variable to the intercept of the measured variables, and that means that you can compare them over time. But if the intercepts of the measured variables drift apart, you can't anchor the means to them any more, because you don't know where they are anchored.
Enough analogies, let's have a concrete example.
Let's say you want to compare depression symptoms in men and women.
So you ask three questions: How many days in the past week have you:
I create a latent variable based on this, and error and loadings look good. Now I want to compare the means of the latent variables, so I fix the male latent mean to zero. I constrain the intercepts of the three measured variables to be equal across groups.
Women and men do not differ on how much they have felt lonely, how much they have felt sad, but then we find that women say that they have cried more than men.
Does that mean that the women have 'more' depression than the men? If we anchor to crying - yes. If we anchor to the other two variables - no. We don't have intercept invariance, and because of that, we can't compare the means of the latent variables.
Another (only slightly different) way to think about it. The intercept of the measured variable is the expected value of the variable if the mean of the factor is equal to zero. The predicted values for the measured variables should be the same between men and women when the values of the factors are equal (that is, when the value of the factors is zero). But the predicted values of the measured variables are not equal when the factors are equal. Some are equal (in our example, 1 and 2), one is not (3).