Analysis of Variance, ANOVA tests whether the means of independent or unrelated groups or treatments are statistically significant. An experiment is assigned factors with different levels, for instance, an experiment with one treatment and a control has one factor (the treatment) and two levels (the treatment and the control). In this case, only one independent variable is manipulated.
ANOVA divides the observed variance into random and systematic variances (Zar, 2009):
Random variance is attributed to experimental manipulation and is the explained from sum of squares between the groups, which is SS (between) or SSX.
Systematic variance is the variance unrelated to X, hence unaccounted for by the sum of squares related to X. It is attributed to sum of squares within the groups; hence SSE or SS (within).
Both random and systematic variances contribute to the total variance, SST. Whereas the random error has no statistical influence on the data, systematic error has strong statistical influence on the data.
F-stat is the ratio between SSX and SSE, and we use the degrees of freedom associated with each variance to find the mean squared error. A high F ratio shows a significant likelihood (and a low probability, p < 0.05), that the observed mean differences were not obtained by chance. Hence the data sampled from different populations had different means, or because the independent treatments on the response variable had a significant effect.
Nevertheless, ANOVA is also described as a method of statistical modeling which is rarely used by researchers. Is there any creative and dynamic way this method can be used to generate new data and make predictions on observations outside the datasets?
Reference
Zar, J. H. (2010). Biostatistical Analysis, Fifth edition ISBN: 0131008463
Isn't ANOVA a linear model so prediction is the same as a linear model? you brought me on an idea in R. To put it to the test if we can predict with an ANOVA:
data(iris)
#ANOVA
print(an
Dear colleagues,
One thing is the model, another thing is ANOVA- Analysis of Variance, which is a methodology. Depending on the model you have different ways to use ANOVA, but one thing you should verify first of all is if your data obey the ANOVA assumptions.
You can use ANOVA for other models than linear.
It's actually the other way around:
Systematic variance is attributed to experimental manipulation and is the explained from sum of squares between the groups, which is SS (between) or SSX.
Random variance is the variance unrelated to X, hence unaccounted for by the sum of squares related to X. It is attributed to sum of squares within the groups; hence SSE or SS (within).
Note that if the experiment is poorly designed then the systematic variation may be attributable to other sources of variability and be totally unrelated to our experimental manipulation. For example if we were silly enough to run all of the first treatment on one day, all of the second treatment on a second day, and all of the third treatment on a third day, we might conclude there were systematic differences but these may be attributable to day-to-day variability rather than our experimental treatments. Worse still we may have genuine differences between the treatments but they are masked by the day-to-day variability. Careful design ensuring your experiments are balanced across the days may allow you to extend the analysis of variance, separating day-to-day, experimental treatment, and random variability.
In the case of one way analysis of variance, the random variability is calculated as the variation around the means, and the systematic variation as the variability of the means around the overall mean. The F test compares the variability in those means relative to that which we would expect given the variability in the data about the means. If the variability in the means is no greater than we would expect given the variability in the data about the means, then we'd expect an F-ratio of 1. Values greater than 1 indicated variation in the means larger than expected. Ho big does the F-value need to be before we get excited? Depends upon the amount of information we have about variability - the degress of freedom.
In what sense is ANOVA a model? In the sense that we're assuming the mean estimates the population mean, the individual means estimate the true group means, the data are independent and identically distributed, and the random or error variability (estimated by the residuals) is (usually) normally distributed.
Alternative error models are available (binomial, Poisson, etc) and these are the basis of the generalized linear models.
Hope this helps.
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