Subjects: Other Disciplines >> Synthetic discipline submitted time 2023-10-09 Cooperative journals: 《心理学报》
Abstract： Latent growth models (LGMs) are a powerful tool for analyzing longitudinal data, and have attracted the attention of scholars in psychology and other social science disciplines. For a latent variable measured by multiple indicators, we can establish both a univariate LGM (also called first-order LGM) based on composite scores and a latent variable LGM (also called second-order LGM) based on indicators. The two model types are special cases of the first-order and second-order factor models respectively. In either case, we need to scale the factors, that is, to specify their origin and unit. Under the condition of strong measurement invariance across time, the estimation of growth parameters in second-order LGMs depends on the scaling method of factors/latent variables. There are three scaling methods: the scaled-indicator method (also called the marker-variable identification method), the effect-coding method (also called the effect-coding identification method), and the latent-standardization method. The existing latent-standardization method depends on the reliability of the scaled-indicator or the composite scores at the first time point. In this paper, we propose an operable latent-standardization method with two steps. In the first step, a CFA with strong measurement invariance is conducted by fixing the mean and variance of the latent variable at the first time point to 0 and 1 respectively. In the second step, estimated loadings in the first step are employed to establish the second-order LGM. If the standardization is based on the scaled-indicator method, the loading of the scaled-indicator is fixed to that obtained in the first step, and the intercept of the scaled-indicator is fixed to the sample mean of the scaled-indicator at the first time point. If the standardization is based on the effect-coding method, the sum of loadings is constrained to the sum of loadings obtained in the first step, and the sum of intercepts is constrained to the sum of the sample mean of all indicators at the first time point. We also propose a first-order LGM standardization procedure based on the composite scores. First, we standardize the composite scores at the first time point, and make the same linear transformation of the composite scores at the other time points. Then we establish the first-order LGM, which is comparable with the second-order LGM scaled by the latent-standardization method. The scaling methods of second-order LGMs and their comparable first-order LGMs are systematically summarized. The comparability is illustrated by modeling the empirical data of a Moral Evasion Questionnaire. For the scaled-indicator method, second-order LGMs and their comparable first-order LGMs are rather different in parameter estimates (especially when the reliability of the scale-indicator is low). For the effect-coding method, second-order LGMs and their comparable first-order LGMs are relatively close in parameter estimates. When the latent variable at the first time point is standardized, the mean of the intercept-factor of the first-order LGM is close to 0 and not statistically significant; so is the mean of the intercept-factor of the second-order LGM through the effect-coding method, but those through two scaled-indicator methods are statistically significant and different from each other. According to our research results, the effect-coding method is recommended to scale and standardize the second-order LGMs, then comparable first-order LGMs are those based on the composite scores and their standardized models. For either the first-order or second-order LGM, the standardized results obtained by modeling composite total scores and composite mean scores are identical.
Subjects: Psychology >> Statistics in Psychology submitted time 2023-03-29
Abstract：
Latent growth models (LGMs) are a powerful tool for analyzing longitudinal data, and have attracted the attention of scholars in psychology and other social science disciplines. For a latent variable measured by multiple indicators, we can establish both a univariate LGM (also called first-order LGM) based on composite scores and a latent variable LGM (also called second-order LGM) based on indicators. The two model types are special cases of the first-order and second-order factor models respectively. In either case, we need to scale the factors, that is, to specify their origin and unit. Under the condition of strong measurement invariance across time, the estimation of growth parameters in second-order LGMs depends on the scaling method of factors/latent variables. There are three scaling methods: the scaled-indicator method (also called the marker-variable identification method), the effect-coding method (also called the effect-coding identification method), and the latent-standardization method.
The existing latent-standardization method depends on the reliability of the scaled-indicator or the composite scores at the first time point. In this paper, we propose an operable latent-standardization method with two steps. In the first step, a CFA with strong measurement invariance is conducted by fixing the mean and variance of the latent variable at the first time point to 0 and 1 respectively. In the second step, estimated loadings in the first step are employed to establish the second-order LGM. If the standardization is based on the scaled-indicator method, the loading of the scaled-indicator is fixed to that obtained in the first step, and the intercept of the scaled-indicator is fixed to the sample mean of the scaled-indicator at the first time point. If the standardization is based on the effect-coding method, the sum of loadings is constrained to the sum of loadings obtained in the first step, and the sum of intercepts is constrained to the sum of the sample mean of all indicators at the first time point. We also propose a first-order LGM standardization procedure based on the composite scores. First, we standardize the composite scores at the first time point, and make the same linear transformation of the composite scores at the other time points. Then we establish the first-order LGM, which is comparable with the second-order LGM scaled by the latent-standardization method.
The scaling methods of second-order LGMs and their comparable first-order LGMs are systematically summarized. The comparability is illustrated by modeling the empirical data of a Moral Evasion Questionnaire. For the scaled-indicator method, second-order LGMs and their comparable first-order LGMs are rather different in parameter estimates (especially when the reliability of the scale-indicator is low). For the effect-coding method, second-order LGMs and their comparable first-order LGMs are relatively close in parameter estimates. When the latent variable at the first time point is standardized, the mean of the intercept-factor of the first-order LGM is close to 0 and not statistically significant; so is the mean of the intercept-factor of the second-order LGM through the effect-coding method, but those through two scaled-indicator methods are statistically significant and different from each other.
According to our research results, the effect-coding method is recommended to scale and standardize the second-order LGMs, then comparable first-order LGMs are those based on the composite scores and their standardized models. For either the first-order or second-order LGM, the standardized results obtained by modeling composite total scores and composite mean scores are identical.
Peer Review Status:
Awaiting Review
Subjects: Psychology >> Social Psychology submitted time 2023-03-28 Cooperative journals: 《心理科学进展》
Abstract： Hypothesis testing is an important part of inferential statistics. Most reported statistical test results are based on the null hypothesis significance test (NHST). In the first two decades of the 21st century, the studies on hypothesis testing and related topics in China’s mainland cover such topics as the deficiency of the null hypothesis significance test, use of P-value, repeatability of psychological research, effect size, power of a statistical test, and equivalence test, among others. This systematic review summarizes the main findings and gives suggestions. NHST has a wide range of applications to a variety of fields, from mathematical statistics to psychology. In the past two decades, Chinese researchers have experienced a process from knowing, using, misunderstanding, understanding, and questioning it, to constantly proposing improvement methods. NHST still occupies an important position in scientific research, despite some shortcomings. When providing statistically significant results, it is recommended to offer precise P-values in order to better evaluate the type I error rate. When one wants to verify is equivalence (or zero effect), a better approach is to set an equivalent boundary value and put the equivalence hypothesis in the position of alternative hypothesis. NHST has been developed into a set of procedures as follows: First, to ensure the power of a statistical test and save costs, one should do a priori power analysis before sampling, and calculate the required sample size. The only exception is questionnaire studies with more than 160 participants which usually do not need such priori power analysis in the traditional statistical analysis. Second, to collect and analyze data, and report NHST results and confidence intervals. Third, to calculate and report the effect size if the results are statistically significant (at this time only the Type Ⅰ error is possible), and draw conclusions based on the magnitude of the effect size. Fourth, to calculate the effect size if the results are not statistically significant (at this time only the Type Ⅱ error is possible), and accept the null hypothesis if the effect size is small. However, a posterior power analysis is required when the effect size is medium or large. If the test power is high, the null hypothesis will be accepted; if the test power is less than 80%, more participants could be added for further analysis. The process of increasing the sample size should be reported clearly, with the final P-value presented and the type I error rate evaluated. Furthermore, the reproducibility crisis of psychological research is partly attributable to NHST. But the reproducibility of scientific research must be strictly defined. Although the failure to replicate a study may result from inaccurate operations and improper methods, it may also be caused by moderating effect. We can't judge the scientificity of a study simply by whether it is replicable. There are three major aspects for expanding the research on the related issues of hypothesis testing. Firstly, the equivalence test has been extended to the evaluation of structural equation models. Second, the analysis of test power has been extended to models other than those in traditional statistics, such as mediation effect models and structural equation models. Third, the effect size has also been extended to models other than those in traditional statistics, and a new R2-type effect size was proposed by using variance decomposition.
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