The likelihood approach is one of several approaches to making inferences from data. The best description and justification for the approach is given by Edwards, A.W.F. (1992) Likelihood, Johns Hopkins Press. Others, such as R. Royall, S. Goodman, Z. Dienes, S. Glover & P. Dixon have subsequently made important contributions.
The likelihood approach focusses on the observed data, using maximum likelihood for estimates, and calculates likelihood ratios for specific parameter values given the collected data. The log of the likelihood is known as the support. When comparing two parameter values (as might be suggested by different hypotheses), the difference in log likelihoods represents the ratio of their likelihoods. Using support has distinct advantages. It represents the weight of evidence with a scale that ranges from positive to negative (indicating support for or against a hypothetical parameter value). Support values obtained from independent studies can simply be added together to give their combined support. Unlike p values, support values are insensitive to transforms.
Support values give the relative strength of evidence for one hypothesis value versus another. They range from negative infinity to positive infinity. There is no specific threshold value (unlike 0.05 in frequentist testing), and values can be rounded to the nearest 1 decimal place, e.g. -2.161 -> -2.2. A support of 0 shows no support for either hypothesis, while a support of 1 represents weak evidence for the first hypothesis versus the second. A support of 2 represents moderate evidence, and is roughly equivalent to the 5% significance level in the frequentist approach. A support of 3 represents strong evidence and 4 represents extremely strong evidence, and so on. Negative values represent the same strength of evidence, except for the second hypothesis value versus the first. Support values for a given likelihood function can be compared with each other simply by subtracting them. In the example below, the support for d versus the 2nd hypothesis can be obtained by subtracting the support for the 2nd hypothesis versus null (4.115) from the support for d versus the null (2.191) giving -1.924, as listed in the 3rd line of output. The support for d versus the observed mean would be 2.191 - 4.352 = -2.161, negative as expected since d lower on the likelihood function curve from the maximum represented by the observed mean (MLE).
The likelihood interval can be obtained for many statistics. This represents the range of values that are consistent with the observed data for a given level of support. For example, the S-2 likelihood interval includes all values which are not more different from the MLE by more than a support value of 2. As noted for the figure below, the S-2 interval is numerically closely equivalent to the 95% confidence interval, as the support of 2 is similar to the 5% significance level. An S-3 likelihood interval would include all values which are not more different from the MLE by more than a support value of 3, and so on. The stronger the interval the wider it will be. The interpretation of the likelihood interval is distinct from a confidence interval. The confidence interval represents the long run probability of capturing the population parameter and may need to be corrected for multiple testing, stopping rule, etc. The likelihood interval is also distinct from the Bayesian credibility interval that represents the subjective probability for a population value occurring within it.
There are few statistical packages that implement the likelihood approach and which calculate support. I have created an R package called likelihoodR which calculates support for a range of statistical analyses: t tests, ANOVA, regression and categorical analyses. In R it can be installed from my GitHub repository using the devtools command:
install.packages("devtools")
devtools::install_github("PeterC-alfaisal/likelihoodR")The functions in the package complement my recent book: Cahusac, P.M.B.
(2020) Evidence-Based Statistics, Wiley
Amazon.co.uk
I would be interested in feedback [email protected] Peter Cahusac
Once installed, full details about each function, including arguments, outputs, and examples, can be obtained in R by typing ?function_name. For example, for the 2 independent samples t test function:
?L_2S_ttestExample for this function (2 independent samples t test), using a variation on Gosset's original additional hours of sleep data
mysample <- c(0.7, -1.6, -0.2, -1.2, -0.1, 3.4, 3.7, 0.8, 0.0, 2.0)
treat <- rep(1:0,each=5)
L_2S_ttest(mysample, treat, veq=0, null=0, d=0.5, alt.2=2, L.int=2, verb = TRUE)## Maximum support for the observed mean 2.46 (dashed line) against the null 0 (black line) = 4.352
## Support for d of 0.5 (0.925, blue line) versus null = 2.191
## Support for d versus 2nd alt Hypothesis 2 (green line) = -1.924
## Support for 2nd alt Hypothesis versus null = 4.115
##
## S-2 likelihood interval (red line) is from 0.99557 to 3.92443
##
## t(6.4) = 2.975, p = 0.02305882, d = 1.33
# One sample and related samples t test
L_ttest(data1, data2, null=0, d=0.5, alt.2=NULL, L.int=2, verb=TRUE)
# Independent samples t test
L_2S_ttest(data, group, veq=0, null=0, d=0.5, alt.2=NULL, L.int=2, verb=TRUE)
# Sample size calculation using the evidential approach for t tests
L_t_test_sample_size(MW = 0.05, sd = 1, d = 1.2, S = 3, paired = FALSE, verb=TRUE)
# One-way independent samples ANOVA
L_1way_ANOVA(data, group, contrast1=NULL, contrast2=NULL, verb=TRUE)
# Two-way independent samples factorial ANOVA
L_2way_Factorial_ANOVA(data, factor1, factor2, contrast1=NULL, contrast2=NULL, verb=TRUE)
# One-way repeated measures ANOVA
L_1way_RM_ANOVA(dat, group, ID, contrast1=NULL, contrast2=NULL, verb=TRUE)
# Correlation
L_corr(xv, yv, null=0, exp.r=NULL, L.int=2, alpha=.05, verb=TRUE)
# Regression, comparing linear, quadratic and cubic fits
L_regress(y, x, verb=TRUE)
# Multiple logistic regression
L_logistic_regress(yv, p1, p2=NULL, p3=NULL, p4=NULL, p5=NULL, p6=NULL, verb=TRUE)
# One-way categorical data, goodness of fit
L_1way_cat(obs, exp.p=NULL, L.int=2, alpha=0.05, toler=0.0001, verb=TRUE)
# Two-way categorical data, tests for interaction and main effects
L_2way_cat(table, verb=TRUE)
# Odds Ratio
L_OR(table, null=1, exp.OR=NULL, L.int=2, alpha=0.05, cc=FALSE, toler=0.0001, verb=TRUE)
# Relative Risk
L_RR(table, null=1, exp.RR=NULL, L.int=2, alpha=0.05, cc=FALSE, toler=0.0001, verb=TRUE)
# Efficacy
L_efficacy(a, n, null=0, exp.eff=NULL, L.int=2, alpha=0.05, toler=0.0001, verb=TRUE)