Parameters to be tested for equality across groups would include the growth factor means and potentially the growth factor (co)variances. To constrain the growth factor parameters, you would specify an equality label in the overall model statement, e.g.,

MODEL:

[i] (1); ! intercept factor mean set equal across groups

[s] (2); ! slope factor mean set equal across groups

i (3); ! intercept factor variance set equal across groups

s (4); ! slope factor variance set equal across groups

i with s (5); ! intercept with slope factor covariance set equal across groups

In addition, you could test whether the first-order factor residual variances are equal; For example, for four time points and four first-order factors F1 through F4:

F1 (6);

F2 (7);

F3 (8);

F4 (9);

My personal opinion Personal autonomy and growth models best

Background: What You’ve Done So Far

You’ve established strict longitudinal invariance in your second-order growth model — excellent! That means:

• Factor loadings (Λ) are equal across time.
• Intercepts (τ) of indicators are equal across time.
• Residual variances (θ) of indicators are equal across time.

This ensures your latent constructs are measured identically over time — a prerequisite for valid growth modeling.

Test Group Invariance for Growth Parameters

Now you want to test whether the growth parameters (intercept, slope, their variances, covariances, etc.) are invariant across groups (e.g., gender). This is sometimes called “structural invariance” or “latent growth invariance.”

You’ll test this in stages — progressively constraining parameters and comparing model fit.

Recommended Testing Sequence (Nested Models)

You’ll compare increasingly constrained models across groups. Here’s the standard sequence:

Since you already have strict longitudinal invariance, you can start from Step 4 (strict measurement invariance across groups) or even Step 5 if you’re confident measurement is invariant across groups as well. But best practice is to test group measurement invariance explicitly.

Mplus Code Template (Second-Order Growth Model by Group)

Assume:

• y11–y14 = indicators at Time 1–4 for construct 1
• y21–y24 = indicators at Time 1–4 for construct 2
• Two 1st-order factors per time point: F1_1, F1_2, F1_3, F1_4 and F2_1, F2_2, F2_3, F2_4
• Second-order factors: I (intercept), S (slope) — regressed on time scores (e.g., 0, 1, 2, 3)

Step 1: Configural Invariance (Baseline)

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TITLE: Configural Invariance - Second-Order Growth Model by Group;

DATA: FILE = yourdata.dat;

VARIABLE:

NAMES = id y11-y14 y21-y24 gender;

USEVARIABLES = y11-y14 y21-y24;

GROUPING = gender (0 = male 1 = female);

ANALYSIS:

ESTIMATOR = MLR;

TYPE = MEANSTRUCTURE;

MODEL:

! First-order factors at each time

F1_1 BY y11* y21; F1_1@1;

F2_1 BY y12* y22; F2_1@1;

F3_1 BY y13* y23; F3_1@1;

F4_1 BY y14* y24; F4_1@1;

! Second-order growth factors

I BY F1_1@1 F2_1@1 F3_1@1 F4_1@1;

S BY F1_1@0 F2_1@1 F3_1@2 F4_1@3;

[I@0 S@0]; ! Optional: set latent means to 0 in configural (or free)

MODEL female:

! Same model, parameters freely estimated

F1_1 BY y11* y21; F1_1@1;

F2_1 BY y12* y22; F2_1@1;

F3_1 BY y13* y23; F3_1@1;

F4_1 BY y14* y24; F4_1@1;

I BY F1_1@1 F2_1@1 F3_1@1 F4_1@1;

S BY F1_1@0 F2_1@1 F3_1@2 F4_1@3;

[I S]; ! Free latent means in one group (female) to identify model

Step 2: Weak Invariance — Constrain Factor Loadings

Add to MODEL: (after baseline):

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MODEL:

! Constrain loadings across groups

F1_1 BY y11 (1)

y21 (2);

F2_1 BY y12 (1)

y22 (2);

F3_1 BY y13 (1)

y23 (2);

F4_1 BY y14 (1)

y24 (2);

I BY F1_1@1 F2_1@1 F3_1@1 F4_1@1;

S BY F1_1@0 F2_1@1 F3_1@2 F4_1@3;

MODEL female:

[I S]; ! Still free latent means

Step 3: Strong Invariance — Constrain Intercepts

Add intercept constraints. You must free latent means in one group (already done).

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MODEL:

! Constrain indicator intercepts

[y11 y12 y13 y14] (3);

[y21 y22 y23 y24] (4);

! Loadings already constrained from Step 2

Step 4: Strict Invariance — Constrain Residual Variances

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MODEL:

y11 y12 y13 y14 (5);

y21 y22 y23 y24 (6);

Step 5: Latent Mean Invariance — Constrain Growth Factor Means

Now constrain means of I and S across groups.

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MODEL:

[I] (7);

[S] (8);

MODEL female:

[I] (7); ! Remove [I S]; and replace with equality constraints

[S] (8);

You must remove [I S]; from MODEL female: — now both groups share the same mean parameters.

Step 6: Latent Variance-Covariance Invariance

Constrain variances and covariance of I and S:

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MODEL:

I (9);

S (10);

I WITH S (11);

MODEL female:

I (9);

S (10);

I WITH S (11);

Model Comparison & Fit Evaluation

Use chi-square difference tests (with MLR: use DIFFTEST or scaled chi-square difference) or information criteria (BIC, AIC) to compare nested models.

Also check:

• ΔCFI < 0.01
• ΔRMSEA < 0.015
• ΔSRMR < 0.01 (for measurement) or < 0.03 (for structural)

If fit doesn’t worsen significantly, invariance holds at that level.

Pro Tips from a Math Ed Researcher

• Always test measurement invariance (Steps 1–4) before comparing growth parameters. Otherwise, group differences may reflect measurement bias, not true growth differences.
• Consider partial invariance if full invariance fails — free non-invariant parameters if theoretically justifiable.
• Label parameters clearly (e.g., (1), (2)) — makes debugging and modification easier.
• Use MODEL CONSTRAINT if you want to test specific contrasts (e.g., difference in slope means).

Recommended Reading

• Meredith, W., & Horn, J. (2001). Factor analysis. In New Handbook of Methods in Applied Psychology.
• Widaman, K. F., & Reise, S. P. (1997). Exploring the measurement invariance of psychological instruments.
• Grimm, L. G., Ram, N., & Estabrook, R. (2017). Growth Modeling: Structural Equation and Multilevel Modeling Approaches. Guilford Press. — Excellent for Mplus code examples.

You’re doing rigorous work — testing group invariance in a second-order growth model is essential for valid group comparisons. Don’t skip steps! Each level of invariance builds confidence that your group differences in growth are real and not artifacts of measurement.