Fixed and Random Models
The classification of factors into fixed or random models is essential in research, and it depends on the researcher's understanding of the field being studied. This classification helps researchers achieve uniform definitions and perceptions.
1. Fixed Model.
The fixed model refers to experiments where the treatment or level of the factor is determined by the researcher before the study begins. The researcher has a reason, usually based on knowledge in his field, to determine that these factors have certain characteristics that distinguish them from other factors. Each level can represent a population hypothesized or imagined by the researcher.
For example, in a study on the influence of Bali bull studs on the birth weight of offspring from uniformly bred females. If four studs are each mated with five uniform female cows, the stud factor could be a fixed or random model.
Bali bull studs are said to be a fixed model, if each stud can be identified as having certain characteristics determined by the researcher before the study. For example, the first stud is 2 years old, the second stud is 2.5 years old, the third stud is 3 years old, and the fourth stud is 3.5 years old. In this case, each stud can represent a set of populations hypothesized or imagined by the researcher.
Another example is when a researcher wants to study the effect of fertilizer variations (factor) on rice crop production. The researcher decides to use three different types of fertilizer: Fertilizer A, Fertilizer B, and Fertilizer C, which have been selected based on certain characteristics. In this context, the type of fertilizer is a fixed model, as the researcher has determined the types of fertilizer to be studied before the experiment begins.
2. Random Model.
Conversely, if a researcher conducts a study on the effect of plant varieties on harvest yields, but the plant varieties used are randomly selected from dozens or hundreds available, then this is an example of a random model. A researcher might randomly choose five varieties from a large population and plant them under identical conditions to see how they grow. In this case, the plant varieties represent a random model, as the researcher did not determine which varieties to use before starting the study.
In a fixed model, the researcher has defined their inference population. Let's say αi (i=1,2,3,...,t) represents the fixed effect of factor A at level I. As αi is considered constant, E(αi) = αi, the true mean of αi.
A factor is considered in the random model if a researcher randomly chooses t levels of a factor (t < T) from the factor's population. In this scenario, repeating to obtain t levels of factor A introduces an element of uncertainty. Let Ai (i=1, 2, 3,...,t) represent the random effect of factor A at level I, the true mean Ai=E(Ai)=0 for all i, as Ai is considered a random variable. Variability in the random model arises not only due to the variability of Ai values but also due to the variability of sample sizes based on the drawing with a choice.
3. Mixed Model.
The mixed model might occur if a researcher investigates the effect of fertilizer and plant varieties on harvest yields. Suppose the researcher has chosen three types of fertilizer (Fertilizer A, Fertilizer B, and Fertilizer C) and randomly selected five rice varieties from a large population. In this case, the fertilizer type represents a fixed model (because the researcher determined the types of fertilizer before the experiment began), while the plant varieties represent a random model (because the researcher randomly chose the varieties). This is an example of a mixed model, where some factors are chosen in a fixed manner, while others are chosen randomly.
The choice between fixed, random, or mixed models has significant consequences for the analytical approach used in the study. Here are some consequences:
- Fixed Model: In a fixed model, the researcher is typically interested in understanding the direct influence of specific levels or categories of the independent variable (factor) on the dependent variable. In this context, analysis of variance (ANOVA) is often used to evaluate the mean differences between groups. The primary goal is to determine if the observed differences between groups significantly differ from what might be expected based on random variation alone. Consequently, if the researcher wants to make inferences or generalizations to other factor levels beyond those chosen in the study, it might be inappropriate because the factor levels were specifically set by the researcher.
- Random Model: In a random model, the researcher is typically interested in variability between levels or categories of the independent variable. An appropriate analysis here might involve ANOVA, but with an understanding that conclusions are about variability in a broader population, not about the specific levels or categories chosen for the study. Consequently, results from a random model analysis can't be directly applied to specific levels or categories of the factor but rather to the entire population.
- Mixed Model: In a mixed model, there's a blend between an interest in specific level or category effects (from the fixed model) and an interest in variability (from the random model). Appropriate analysis might involve techniques like mixed analysis of variance or mixed linear models. Consequently, the study can become more complex and might require a deeper understanding of statistics and modeling.
In the context of experimental design, the choice of a fixed, random, or mixed model will significantly influence how data is analyzed and how the results are interpreted. Therefore, it's crucial for researchers to consider the research objectives, desired population, and the type of inference wanted before deciding on the appropriate model to use.
Calculations with Data Processing Application
SmartstatXL (Excel Add-In)
Analysis of Variance calculations for various Experimental Designs and further tests (LSD, Tukey's HSD, Scheffé's test, Duncan, SNK, Dunnet, REGWQ, Scott Knott) using SmartstatXL can be learned from the following link: SmartstatXL Add-In Documentation
