In econometrics and statistics, a fixed effects model is a statistical model that represents the observed quantities in terms of explanatory variables that are treated as if the quantities were non-random. This is in contrast to random effects models and mixed models in which either all or some of the explanatory variables are treated as if they arise from random causes. Contrast this to the biostatistics definitions, as biostatisticians use “fixed” and “random” effects to respectively refer to the population-average and subject-specific effects (and where the latter are generally assumed to be unknown, latent variables). Often the same structure of model, which is usually a linear regression model, can be treated as any of the three types depending on the analyst’s viewpoint, although there may be a natural choice in any given situation.
While a random effect(s) model, also called a variance components model, is a kind of hierarchical linear model. It assumes that the dataset being analysed consists of a hierarchy of different populations whose differences relate to that hierarchy. In econometrics, random effects models are used in the analysis of hierarchical or panel data when one assumes no fixed effects (it allows for individual effects). The random effects model is a special case of the fixed effects model. Contrast this to the biostatistics definitions, as biostatisticians use “fixed” and “random” effects to respectively refer to the population-average and subject-specific effects (and where the latter are generally assumed to be unknown, latent variables).
Mathematically, the difference between fixed and random effects modeling is that the latter uses a multilevel approach to estimate the variation in a response across multiple groups of observations.* In practice this means that, if you have few data points in a group, the group’s effect estimate will be based partially on the more abundant data from other groups. This is known as partial pooling, and it’s a nice compromise between completely pooling all groups, which could mask group-level variation, and treating all groups completely separately, which could give poor estimates for low-sample groups.
An example should help prime your intuition. Suppose you want to estimate average US household income by ZIP code. You have a large dataset containing observations of households’ incomes and ZIP codes. Some ZIP codes are well represented in the dataset, but others have only a couple households.
For your initial model you would most likely take the mean income in each ZIP. This will work well when you have lots of data for a ZIP, but the estimates for your poorly sampled ZIPs will suffer from high variance. You can mitigate this by using a shrinkage estimator (aka partial pooling), which will push extreme values towards the mean income across all ZIP codes.

20 Fixed effect: Something the experimenter directly manipulates and is often repeatable, e.g., drug administration – one group gets drug, one group gets placebo.
Random effect: Source of random variation / experimental units e.g., individuals drawn (at random) from a population for a clinical trial. Random effects estimates the variability

A method based on chance alone by which study participants are assigned to a treatment group. Randomization minimizes the differences among groups by equally distributing people with particular characteristics among all the trial arms. The researchers do not know which treatment is better. From what is known at the time, any one of the treatments chosen could be of benefit to the participant.
Randomization is a technique used to balance the effect of extraneous or uncontrollable conditions that can impact the results of an experiment. For example, ambient temperature, humidity, raw materials, or operators can change during an experiment and inadvertently affect test results. By randomizing the order in which experimental runs are done, you reduce the chance that differences in experimental materials or conditions strongly bias results. Randomization also lets you estimate the inherent variation in materials and conditions so that you can make valid statistical inferences based on the data from your experiment.
Suppose you work for an offset printing company interested in maximizing the effectiveness of their bookbinding technique. You can control factors such as glue temperature, paper type, and cooling time. However, you cannot control humidity, which can affect how quickly the glue sets. Or, perhaps there are other “unknowns” that cannot be easily controlled or measured. For example, the bookbinding machine might not be applying consistent pressure.
When you create a designed experiment, Minitab automatically randomizes the run order, or ordered sequence of the factor combinations, of the design. For example, a 2-level full factorial design based on the bookbinding example yields the following results, (which will vary because of randomization):
C1 C2 C3 C4 C5 C6 C7
StdOrder RunOrder CenterPt Blocks Glue Temp Paper Type Cooling Time
5 1 1 1 250 Gloss 24
1 2 1 1 250 Gloss 12
7 3 1 1 250 Matte 24
6 4 1 1 350 Gloss 24
2 5 1 1 350 Gloss 12
3 6 1 1 250 Matte 12
4 7 1 1 350 Matte 12
8 8 1 1 350 Matte 24
Minitab reserves and names C1 (StdOrder) and C2 (RunOrder) to store the standard order and run order, respectively.
• StdOrder shows what the order of the runs in the experiment would be if the experiment was done in standard order, or Yates order.
• RunOrder shows what the order of the runs in the experiment would be in random order. This is the order you should follow when you run the experiment.
If you do not randomize, the run order and the standard order are the same. There may be situations when randomization leads to an undesirable run order. For instance, in industrial applications, it may be difficult or expensive to change factor levels. Or, after factor levels are changed, it may take a long time for the system to return to a steady state. Under these conditions, you may not want to randomize. Alternatively, you may want to randomize with a split-plot design in order to minimize the level changes.
If you want to re-create a randomized design with the same run order, you can choose a base for the random number generator. Then, when you want to re-create the design, you use the same base. Randomization is not haphazard. Instead, a random process is a sequence of random variables describing a process whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions. For example, a random sample of individuals from a population refers to a sample where every individual has a known probability of being sampled. This would be contrasted with nonprobability sampling where arbitrary individuals are selected.

Flowchart of four phases (enrollment, intervention allocation, follow-up, and data analysis) of a parallel randomized trial of two groups, modified from the CONSORT 2010 Statement
The use of chance alone to assign the participants in an experiment or trial to different groups in order to fairly compare the outcomes with different treatments. Randomization is an important feature of experimental design.
Researchers in life science research demand randomization for several reasons. First, subjects in various groups should not differ in any systematic way. In a clinical research, if treatment groups are systematically different, research results will be biased. Suppose that subjects are assigned to control and treatment groups in a study examining the efficacy of a surgical intervention. If a greater proportion of older subjects are assigned to the treatment group, then the outcome of the surgical intervention may be influenced by this imbalance. The effects of the treatment would be indistinguishable from the influence of the imbalance of covariates, thereby requiring the researcher to control for the covariates in the analysis to obtain an unbiased result.
Second, proper randomization ensures no a priori knowledge of group assignment (i.e., allocation concealment). That is, researchers, subject or patients or participants, and others should not know to which group the subject will be assigned. Knowledge of group assignment creates a layer of potential selection bias that may taint the data. Schul and Grimes stated that trials with inadequate or unclear randomization tended to overestimate treatment effects up to 40% compared with those that used proper randomization. The outcome of the research can be negatively influenced by this inadequate randomization.
Statistical techniques such as analysis of covariance (ANCOVA), multivariate ANCOVA, or both, are often used to adjust for covariate imbalance in the analysis stage of the clinical research. However, the interpretation of this post adjustment approach is often difficult because imbalance of covariates frequently leads to unanticipated interaction effects, such as unequal slopes among subgroups of covariates. One of the critical assumptions in ANCOVA is that the slopes of regression lines are the same for each group of covariates. The adjustment needed for each covariate group may vary, which is problematic because ANCOVA uses the average slope across the groups to adjust the outcome variable. Thus, the ideal way of balancing covariates among groups is to apply sound randomization in the design stage of a clinical research (before the adjustment procedure) instead of post data collection. In such instances, random assignment is necessary and guarantees validity for statistical tests of significance that are used to compare treatments.

The benefits of randomization are numerous. It ensures against the accidental bias in the experiment and produces comparable groups in all the respect except the intervention each group received. The purpose of this paper is to introduce the randomization, including concept and significance and to review several randomization techniques to guide the researchers and practitioners to better design their randomized clinical trials. Use of online randomization was effectively demonstrated in this article for benefit of researchers. Simple randomization works well for the large clinical trails (n>100) and for small to moderate clinical trials (n<100) without covariates, use of block randomization helps to achieve the balance. For small to moderate size clinical trials with several prognostic factors or covariates, the adaptive randomization method could be more useful in providing a means to achieve treatment balance.

1. Frane JW. A method of biased coin randomization, its implementation and validation. Drug Inf J. 1998;32:423–32.
2. Altaman DG, Bland JM. How to use randomize. BMJ. 1999;319:703–4. [PMC free article] [PubMed]
3. Altaman DG, Bland JM. Statistics notes. Treatment allocation in controlled trails: Why randomize? BMJ. 1999;318:1209. [PMC free article] [PubMed]
4. R development Core Team. An Introduction to R 2004. (First Edition) ISBN 0954161742.
5. SAS/Stat User's Guide, version 9.2. Cary, NC: SAS Institute Inc; 2009. SAS institute Inc.
6. Domanski M, Mckinla . A Handbook for the 21st century. Philadephia, PA: Wolters Kulwer; 2009. Successful randomized trails.
7. Kalish LA, Begg GB. Treatment allocation methods in clinical trials a review. Stat Med. 1985;4:129–44. [PubMed]
8. Fleiss JL, Levin B, Park MC. A statistical Methods for Rates and Proportion. 3rd ed. Hoboken NJ: John Wiley and Sons; 2003. How to randomize.
9. Schul KF, Grimes DA. Allocation concealment in randomized trials: Defending against deciphering. Lancet. 2002;359:614–8. [PubMed]

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