3 Sample Pages from Langley's "UNDERSTANDING EASY-STATS". Best print 12 characters/inch (Elite) with 1 inch left margin. ------------------------------------------------------------------------------- ANOVA 1-Way But how could "small" or "big" be assessed objectively? Fisher saw that since the variance of means is defined by Var(XBAR) = Var(X) / n, cross-multiplying Var(XBAR) by n will give another statistic which will be an UNBIASED ESTIMATOR OF THE POPULATION VARIANCE. He called this the BETWEEN-SAMPLES MEAN SQUARE --- MS_between = SS_between / df_between where SS_between = n * ä (XBAR_i - XDOUBLEBAR)ý, & df_between = g - 1. F TEST ------ It is important to realize that Between and Within MS's are INDEPENDENT of one another. The spread between means needn't affect the spread within samples, and vice versa. This fact permitted the development of the F TEST to compare these 2 estimates of some åý (common to the populations from which the samples have been drawn), using the famous formula --- F(df_betw, df_within) = MS_betw / MS_within which tests --- H0: åý_between = åý_within H1: åý_between > åý_within (1-tail) Note (1): These hypotheses are exactly equivalent to those expressed in terms of æ_i, at the top of this section. Note (2): This is a 1-tail test because åý_between must be abnormally large if there is a real difference between the population means. åý_between can only become very small if the means are very close together. Note (3): F Tables for interpreting the variance ratio values only show the right hand tail (F values > 1), since ANOVA is their main use. [See VARIANCE RATIO TEST for their use as 2-tailed tests.] Note (4): Big values of F are produced by much spread between sample means, and will reject H0. Note (5): Values of F < 1 occur if the sample means are closer together than expected with random sampling. This will happen sometimes by chance when H0 is true. But don't guess. See the "F TABLES" notes herein for how to find its probability --- if P > 5% accept H0, otherwise suspect some violation of assumptions such as non-random sampling or unequal population variances. [Ref: Bennett & Franklin 7.25] Note (6): See ANOVA ASSUMPTIONS, which tells when you can trust this F Test. TOTAL MEAN SQUARE ----------------- A third variance can also be computed from multiple samples, namely, the TOTAL MEAN SQUARE. This is a measure of the spread of all the sampled measurements around their grand mean --- MS_total = SS_total / df_total where SS_total = ä (X - XDOUBLEBAR)ý, & df_total = N - 1, & N = än_i = total number of measurements in all g samples. All these SS's, df's, and MS's are displayed in an ANOVA TABLE, together with the F Test. The MS_total is not independent of the other two MS's, so isn't used for testing these hypotheses. ------------------------------------------------------------------------------- UNDERSTANDING EASY-STATS CORRELATION, Grouped Data A scattergram of these figures would be like this --- Aggression Score 50+ Y | o 40+ o | o 30+ 2 | 20+ o | o 10+-------+-------+-------+-------+ 0 1 2 3 4 Birth Order, X To get Pearson's r, you could enter these pairs into CORRELATIONS (VARIOUS), but it will be quicker, and you'll get a LINEARITY TEST of the relationship, if you use our REGROUP program to regroup the pairs by the X-variable, then look on X as a sample ID, and enter the Y-values into the 1-WAY ANOVA program, thus --- Sample # Scores 1 20 15 2 39 28 29 4 46 37 Choose a Weighted Means Analysis, and when asked --- "Are the levels of Factor `A' Quantitative?" - answer Y for yes, then "Enter `E' if Equally spaced, otherwise enter their 3 values in free format:" - enter 1 2 4 to suit the present case. For a more detailed analysis of relationships with repeated X's, use our REGRESSION program. CORRELATION, PARTIAL ---------------------- "Partialling" was introduced by Yule (1897) to correct an observed correlation between 2 variables for the disturbing influence of other variables (which are then said to be "partialled out" of the main correlation). E.g. the correlation between reading and writing computed from a random sample of children of various ages could be wrong because the relationship may depend in part on age. Instead of restricting the sample to children of the same age ("experimental control"), we can statistically "partial out" the effect of age on the reading and writing scores. This could be done by using each child's reading and writing DEVIATE SCORE from the mean of his/her age group. The unadulterated correlation could then be obtained by correlating these deviate scores, from which the influence of age has been purged. In practice, alternative formulations, based on the correlations between all possible pairs of variables, are used. The net result will be AS THOUGH the childrens' ages had been constant in the sample. ------------------------------------------------------------------------------- UNDERSTANDING EASY-STATS NON-PARAMETRIC TESTS It must be stressed that parametric tests (e.g. Student's t) have been formulated to apply to random samples from populations with certain characteristics (e.g. Normal Distribution). You must not expect them to give true answers if applied to data from populations which don't conform to such specifications (e.g. if the population is Lognormal when a test assumes a symmetrical distribution). Don't take this too lightly --- it is my experience that about 50% of biological measurements are Lognormal. Non-parametric tests are generally safe to use when analysing measurements and you're not sure about their scale &/or population features. Accordingly, they have much to recommend for novices. But let's face it, if the assumptions for a parametric test are met, the use of a parametric test will usually give a somewhat stronger test (i.e. smaller P-values) than a non-parametric alternative. And furthermore, the mathematical restrictions of ranks and counts is why they cannot be used for sophisticated analyses like ANCOVA or multiple regression. [Ref: Bradley Chap 2] NORMALITY TESTS ----------------- The EASY-STATS Descriptive Statistics provides the following tests to assess whether your sample measurements are likely to derive from a Normally Distributed population or not --- HISTOGRAM of Z SCORES. THOMPSON & GRUBBS' TEST [see OUTLIERS]. SKEWNESS COEFFICIENT & KURTOSIS COEFFICIENT. RANGE/SD RATIO [see OUTLIERS]. Other programs in this package also use these tests when appropriate. ODDS RATIO ------------ See ASSOCIATION, STRENGTH OF. OUTLIERS ---------- Outliers are measurements which differ considerably from the rest of the values in your sample. Outliers may be extreme-but-valid members of the parent population (in which case discarding them would bias results), or they may be truly illegal values (in which case results will be biased unless you do discard them). If the smallest or largest value in the sample can be traced to a clerical or instrumental error, discard it and re-test the remaining values. If the parent population is expected to have a Normal Distribution, outliers should be detected by any NORMALITY TEST (e.g. below), though these tests vary in the features to which they are most sensitive. However, if you are unsure about the distribution the parent population, you should analyse the data WITH and WITHOUT the suspect outlier (and discard the whole sample and start afresh if the outcomes differ importantly --- you mustn't trust a conclusion hanging on 1 suspicious value). [Ref: Kruskal 1960]