Random effects meta regression model

The analyst uses the Meta Regression procedure to fit a random effects meta-regression model using the cleaned data from the glucose.sav sample file and the following syntax:

META REGRESSION LnRiskRatio with length
/DATA VAR=VARLnRiskRatio
/CRITERIA CILEVEL=95 CLASSMISSING=EXCLUDE MAXITER=200 MAXSTEP=100 CONVERGENCE=0.000001
/INFERENCE MODEL=RANDOM INTERCEPT=INCLUDE DISTRIBUTION=NORMAL ESTIMATE=ML ADJUSTSE=NONE
/PRINT COEFF_TEST PARAMETER.

The analyst wants to draw statistical inference about the effect of the oral medicine by computing the effect size, log risk-ratio, and corresponding variance based on four variables, treatSuccess, treatFailure, controlSuccess, and controlFailure. The analyst runs the following syntax to compute the log risk-ratio and corresponding variance.

compute LnRiskRatio=LN(treatsuccess)-LN(treatSuccess+treatFailure)-LN(controlSuccess)+LN(controlSuccess+controlFailure).
compute VarLnRiskRatio=1/treatSuccess+1/controlSuccess-1/(treatSuccess+treatFailure)-1/(controlSuccess+controlFailure).
EXECUTE.

The following section details how to compute the log risk-ratio and corresponding variance through the Compute Variable feature.

Computing the log risk-ratio and variance

  1. Open the glucose_length.sav sample file.
  2. To compute the log risk-ratio, from the menus choose:

    Transform > Compute Variable...

    Figure 1. Compute Variable dialog
    Compute Variable dialog that defines the risk-ratio variable
  3. Enter LnRiskRatio in the Target Variable field, and paste the following equation into the Numeric Expression field.
    LN(treatsuccess)-LN(treatSuccess+treatFailure)-LN(controlSuccess)+LN(controlSuccess+controlFailure)
  4. Click OK. The new variable LnRiskRatio displays in Data Editor.
    Figure 2. Data Editor dialog with LnRiskRatio highlighted
    Data Editor dialog with LnRiskRatio highlighted
  5. To compute the log risk-ratio variance, from the menus choose:

    Transform > Compute Variable...

    Figure 3. Compute Variable dialog
    Compute Variable dialog that defines the risk-ratio variance variable
  6. Click Reset to clear the defined numeric expression.
  7. Enter VarLnRiskRatio in the Target Variable field, and paste the following equation into the Numeric Expression field.
    1/treatSuccess+1/controlSuccess-1/(treatSuccess+treatFailure)-1/(controlSuccess+controlFailure)
  8. Click OK. The new variable VarLnRiskRatio displays in Data Editor.
    Figure 4. Data Editor dialog with VarLnRiskRatio highlighted
    Data Editor dialog with VarLnRiskRatio highlighted

The following section details how to draw statistical inference about the effect through the Meta-Analysis: Regression procedure.

Running the analysis

  1. To run a Meta-Analysis: Regression analysis, from the menus choose:

    Analyze > Meta Analysis > Meta Regression

    Figure 5. Meta-Analysis: Regression main dialog
    Meta-Analysis: Regression dialog with LnRiskRatio selected as the Effect Size variable, VarLnRiskRatio selected as the Variance variable, and length selected as the Covariate variable.
  2. Select LnRiskRatio as the Effect Size variable, select the Variance setting and VarLnRiskRatio as the variable that specifies the variance that is converted to the weight, and length in days [length] as a Covariate(s) variable.
  3. Click Criteria.
    Figure 6. Criteria dialog
    Meta-Analysis: Regression Criteria dialog
  4. Enter 200 in the Maximum Iterations field.
  5. Enter 100 in the Maximum step-halvings field.
  6. Click Continue.
  7. Click Inference.
    Figure 7. Inference dialog
    Meta-Analysis: Regression Inference dialog
  8. Select the Include intercept in regression, Maximum likelihood (ML), and No adjustment settings.

    The Maximum likelihood (ML) setting builds the random-effects meta-regression model with the ML method for estimating the between-study variance.

  9. Click Continue.
  10. Click Print.
    Figure 8. Print dialog
    Meta-Analysis: Regression Print dialog
  11. Select the Model coefficient test and Display exponentiated statistics settings.
  12. Click Continue.
  13. Click OK.

Model Coefficient Test table

Figure 9. Model Coefficient Test table
Model Coefficient Test table

The Model Coefficient Test table displays a test that all coefficients other than the intercept are equal to 0 based on the Wald Chi-squared test. The Wald Chi-squared test statistic is 17.631 and the significance level value is smaller than 0.001, which means the null hypothesis that all coefficients other than the intercept are equal to 0 will be rejected at the significant level 0.05. In other words, the alternative hypothesis that all coefficients are statistically significant different from 0 should be accepted.

Parameter Estimates table

Figure 10. Parameter Estimates table
Parameter Estimates table

The Parameter Estimates table shows the coefficient of the variable length in days as 0.117, which means that every day of length in days corresponds to an increase of 0.117 units in log risk-ratio. The significance value of the test for length in days is smaller than 0.001, which suggests that the coefficient of length in days is significantly different from zero at the significance level 0.05.

Note: Because there is only one moderator (length in days), the z test is equivalent to the Wald Chi-square test in the Model Coefficient Test table (the Wald Chi-square value equals to squared z value).

Test of Residual Homogeneity table

Figure 11. Test of Residual Homogeneity table
Test of Residual Homogeneity table

The Test of Residual Homogeneity table reports the hypothesis test that the between-study variance (tau-squared) is equal to 0. The Chi-square test statistic is 124.551 and the significance value is 0.000, which suggest the presence of heterogeneity among the residuals.

Residual Heterogeneity table

Figure 12. Residual Heterogeneity table
Residual Heterogeneity table

The Residual Heterogeneity table shows that the between-study variance is estimated as 0.275. The heterogeneity statistics (I-squared) is 88.4%, which means that 88.4% of the variability in the residuals is attributed to the between-study variation, whereas 11.6% is attributed to the within-study variation. The R-squared value is 61.6%, which means about 61.6% of the between-study variance is explained by the moderator (length in days).

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