Joseph Lee Rodgers

Thirteen Ways to Look at the Correlation Coefficient

Abstract In 1885, Sir Francis Galton first defined the term “regression” and completed the theory of bivariate correlation. A decade later, Karl Pearson developed the index that we still use to measure correlation, Pearsons r. Our article is written in recognition of the 100th anniversary of Galtons first discussion of regression and correlation. We begin with a brief history. Then we present 13 different formulas, each of which represents a different computational and conceptual definition of r. Each formula suggests a different way of thinking about this index, from algebraic, geometric, and trigonometric settings. We show that Pearsons r (or simple functions of r) may variously be thought of as a special type of mean, a special type of variance, the ratio of two means, the ratio of two variances, the slope of a line, the cosine of an angle, and the tangent to an ellipse, and may be looked at from several other interesting perspectives.

Resolving the Debate Over Birth Order, Family Size, and Intelligence

Hundreds of research articles have addressed the relationship between birth order and intelligence. Virtually all have used cross-sectional data, which are fundamentally flawed in the assessment of within-family (including birth order) processes. Although within-family models have been based on patterns in cross-sectional data, a number of equally plausible between-family explanations also exist. Within-family (preferably intact-family) data are prerequisite for separating within- and between-family causal processes. This observation reframes an old issue in a way that can be easily addressed by studying graphical patterns. Sibling data from the National Longitudinal Survey of Youth are evaluated, and the results are compared with those from other studies using within-family data. It appears that although low-IQ parents have been making large families, large families do not make low-IQ children in modern U.S. society. The apparent relation between birth order and intelligence has been a methodological illusion.

The Epistemology of Mathematical and Statistical Modeling: A Quiet Methodological Revolution.

A quiet methodological revolution, a modeling revolution, has occurred over the past several decades, almost without discussion. In contrast, the 20th century ended with contentious argument over the utility of null hypothesis significance testing (NHST). The NHST controversy may have been at least partially irrelevant, because in certain ways the modeling revolution obviated the NHST argument. I begin with a history of NHST and modeling and their relation to one another. Next, I define and illustrate principles involved in developing and evaluating mathematical models. Following, I discuss the difference between using statistical procedures within a rule-based framework and building mathematical models from a scientific epistemology. Only the former is treated carefully in most psychology graduate training. The pedagogical implications of this imbalance and the revised pedagogy required to account for the modeling revolution are described. To conclude, I discuss how attention to modeling implies shifting statistical practice in certain progressive ways. The epistemological basis of statistics has moved away from being a set of procedures, applied mechanistically, and moved toward building and evaluating statistical and scientific models.

The Bootstrap, the Jackknife, and the Randomization Test: A Sampling Taxonomy.

A simple sampling taxonomy is defined that shows the differences between and relationships among the bootstrap, the jackknife, and the randomization test. Each method has as its goal the creation of an empirical sampling distribution that can be used to test statistical hypotheses, estimate standard errors, and/or create confidence intervals. Distinctions between the methods can be made based on the sampling approach (with replacement versus without replacement) and the sample size (replacing the whole original sample versus replacing a subset of the original sample). The taxonomy is useful for teaching the goals and purposes of resampling schemes. An extension of the taxonomy implies other possible resampling approaches that have not previously been considered. Univariate and multivariate examples are presented.

Did fertility go up after the oklahoma city bombing? An analysis of births in metropolitan counties in Oklahoma, 1990–1999

Political and sociocultural events (e.g., Brown v. Board of Education in 1954 and the German reunification in 1989) and natural disasters (e.g., Hurricane Hugo in 1989) can affect fertility. In our research, we addressed the question of whether the Oklahoma City bombing in April 1995, a man-made disaster, influenced fertility patterns in Oklahoma. We defined three theoretical orientations—replacement theory, community influence theory, and terror management theory—that motivate a general expectation of birth increases, with different predictions emerging from time and geographic considerations. We used two different empirical methodologies. First, we fitted dummy-variable regression models to monthly birth data from 1990 to 1999 in metropolitan counties. We used birth counts to frame the problem and general fertility rates to address the problem formally. These analyses were organized within two design structures: a control-group interrupted time-series design and a difference-in-differences design. In these analyses, Oklahoma County showed an interpretable, consistent, and significant increase in births. Second, we used graphical smoothing models to display these effects visually. In combination, these methods provide compelling support for a fertility response to the Oklahoma City bombing. Certain parts of each theory helped us organize and understand the pattern of results.

The season-of-birth paradox

n There is a seasonal pattern to births in the US: they peak in August-September, peak less strongly in December-February, and show a deep valley in April. What makes this pattern paradoxical is that, as was shown by a survey of 235 undergraduates, the preferred birth months are April-May, and the least desired months are July-August and December-January. 3 hypotheses have been put forward to explain this paradox. The Bad Data Hypothesis holds that college students are not a proper model for the decision making patterns of actual married couples. The Biological Domination Hypothesis holds that birth patterns are not under volitional control, but are determined by hormones, the pineal gland, weather patterns and fetal mortality patterns. The Misinformed Reproducer Hypothesis holds that couples underestimate the lag time between discontinuing contraception and becoming pregnant. To test this hypothesis, married women, aged 15-44, whose contraceptive and pregnancy histories were known, were selected from the National Survey of Family Growth Cycles for births in 1973-75 and 1979-81. Of these, 1271 women stopped contracepting in order to get pregnant and remembered when they had stopped contracepting. The months of stopping contraception were statistically averaged over the 6 years, and in each year there was a valley in February, March or April; and in 5 of the 6 years there was a peak in June. An 11-month dummy variable regression model was used to test the reliability of these patterns for statistical significance. The analysis showed that couples stopped using contraceptives on the assumption that pregnancy would ensue almost immediately. 10 months from February-April are December-February, and 10 months from June is April. If a 5-month pregnancy lag were added to the 9-month gestation, then births would peak in May-August and the valley would be in January-March. This pattern is still 2 months off from the actual birth distribution; however, the retrospective data probably underestimate the real pregnancy lag.n

Does Having Boys or Girls Run in the Family

Several years ago, the first author’s sisternoted, without even a trace of doubt,that “Rodgers men produce boys.” Inever take those kinds of statementsfrom my sister lightly. She is a journalistwho writes popular articles about familyand reproduction, and her husband is anM.D. who specializes in infertility. Fur-ther, she had the data on her side. At thatpoint, the eight Rodgers men with chil-dren from the past four generations hadcollaborated with eight different womenin the production of 24 biological chil-dren; 21 were boys and 3 were girls.Although she had not run the analysis,my sister’s statistical intuition was excel-lent. In the United States, approximately51% of the babies born are males. Undera binomial model of sex outcomes, hav-ing three or fewer boys out of 24 chil-dren would happen by chance aroundtwice in 10,000 families.Like my sister, many mothers andfathers believe that a tendency to haveboys or girls runs in a family. Informally,we have noted that belief among manyof our friends. Pregnant women appearparticularly interested in and amenableto the notion that sex composition runsin the family. But research in the statis-tical and cognitive psychology literaturesuggests that humans are notoriouslybad at distinguishing systematic pat-terns from random patterns. Even if thesex selection process is purely bychance, some parents will have all boysin families of size 1, 2, 3, or even 10 or12. For example, mothers of four chil-dren who have all boys or all girls mustnaturally wonder if something system-atic contributed to their “unusual sexcomposition.” Yet, around 1/8th of allfour-child families are expected to be asame-sex family under a chance model,not an especially unusual occurrence.Yet, even parents with Ph.D.’s in statis-tics must be inclined to wonder whethersuch extreme outcomes are caused byan unusual chance event or by a bias incertain fathers or mothers (or combina-tions) to produce one or the other sex.Some dice really are loaded.Many factors have been identifiedthat can potentially affect the human sexratio at birth. A 1972 paper by MichaelTeitelbaum accounted for around 30such influences, including drinkingwater, coital rates, parental age, parentalsocioeconomic status, birth order, andeven some societal-level influences likewars and environmental pathogens. A1997 study in

Corrections For Spurious Influences On Correlations Between Mmpi Scales

Correlations between measures containing common elements have a spurious component due to the overlapping elements. For example, personality scales such as the MMPI and the CPI have subscales with overlapping items, so correlations between these subscales are spuriously influenced by this overlap. Methods for statistical and experimental adjustment of these spuriously influenced correlations are reviewed and compared, but are rejected in favor of Bashaw and Andersons (1967) reliability solution. To illustrate the proposed correction and to show the potential magnitude of this spurious influence, the original Minnesota normals data on nine clinical subscales of the MMPI were analyzed. A comparison of the unadjusted and adjusted values showed that 32 of the 36 correlations were changed, six of them were radically changed, and in two cases the sign of the correlation was reversed. A principal components analysis was used to investigate changes in the correlation structure.

The Cross-Generational Mother–Daughter–Aunt–Niece Design: Establishing Validity of the MDAN Design with NLSY Fertility Variables

Using National Longitudinal Survey of Youth (NLSY) fertility variables, we introduce and illustrate a new genetically-informative design. First, we develop a kinship linking algorithm, using the NLSY79 and the NLSY-Children data to link mothers to daughters and aunts to nieces. Then we construct mother–daughter correlations to compare to aunt–niece correlations, an MDAN design, within the context of the quantitative genetic model. The results of our empirical illustration, which uses DF Analysis and generalized estimation equations (GEE) to estimate biometrical parameters from NLSY79 sister–sister pairs and their children in the NLSY-Children dataset, provide both face validity and concurrent validity in support of the efficacy of the design. We describe extensions of the MDAN design. Compared to the typical within-generational design used in most behavior genetic research, the cross-generational feature of this design has certain advantages and interesting features. In particular, we note that the equal environment assumption of the traditional biometrical model shifts in the context of a cross-generational design. These shifts raise questions and provide motivation for future research using the MDAN and other cross-generational designs.

Seriation and multidimensional scaling: A data analysis approach to scaling asymmetric proximity matrices

A number of model-based scaling methods have been developed that apply to asymmetric proximity matrices. A flexible data analysis approach is pro posed that combines two psychometric procedures— seriation and multidimensional scaling (MDS). The method uses seriation to define an empirical order ing of the stimuli, and then uses MDS to scale the two separate triangles of the proximity matrix defined by this ordering. The MDS solution con tains directed distances, which define an extra dimension that would not otherwise be portrayed, because the dimension comes from relations between the two triangles rather than within triangles. The method is particularly appropriate for the analysis of proximities containing temporal information. A major difficulty is the computa tional intensity of existing seriation algorithms, which is handled by defining a nonmetric seriation algorithm that requires only one complete itera tion. The procedure is illustrated using a matrix of co-citations between recent presidents of the Psychometric Society.