When engaged in interval estimation, a researcher will (1) select a level of confi-dence (e.g., 95 percent), (2) analyze the sample data, (3) extract a number out of a statistical table, and (4) scientifically build an interval that surrounds the sample sta-tistic. After completing these four steps, the researcher makes an educated guess as to the unknown value of the population parameter. In making this guess, the re-searcher ends up saying, “My data-based interval extends from ______ to ______, and the chances are ______ out of 100 that this interval is one of the many possi-ble intervals (each based on a different sample) that would, in fact, contain the pa-rameter between the interval limits.”
A second form of estimation is called point estimation, and here again, an ed-ucated guess is made, on the basis of sample data, as to the unknown value of the pop-ulation parameter. With this second kind of estimation, however, the activities and thinking of the researcher are much simpler. With point estimation, no level of confi-dence must be selected, no statistical table must be consulted, and no interval must be created. Instead, the researcher simply computes the statistic on the basis of the sam-ple data and then posits that the unknown value of the population parameter is the same as the data-based number. Thus, the researcher who uses this guessing technique ends up saying, “Because the sample-based statistic turned out equal to ______, my best guess is that the value of the parameter is also equal to that particular value.”
Point estimation, of course, is likely to produce statements that are incorrect.
Because of the great likelihood of sampling error, the value of the statistic rarely matches the value of the parameter. For this reason, interval estimation is generally considered to represent a more logical way of making educated guesses as to para-meter values than is point estimation.
Despite the fact that point estimation disregards the notion of sampling error, many researchers can be seen making pinpoint guesses as to parameter values.
Consider, for example, Excerpt 6.8. In the research report from which this excerpt was drawn, the researchers looked at the relationship between life satisfaction and tolerance.
In Excerpt 6.8, two correlation coefficients are presented. Each of these rs is a point estimate, even though this label does not appear in the excerpt (or in the full research report). Despite the fact that these correlations were based on large samples—approximately 1,000 adults in each of 150 countries who participated in the Gallop Organization’s World Poll—the rs reported in Excerpt 6.8 are still point estimates. If different people had been interviewed, the correlations most likely would have been at least slightly different. Because the phrase point estimate or simply the word estimate is nowhere to be seen, I am afraid readers of this research report might mistakenly consider these correlations to be population parameters rather than sample statistics.
In Excerpt 6.9, we see an example of researchers pointing out that their point estimates were just that, point estimates. The final sentence of this excerpt provides a warning to readers of the research report not to interpret the numbers EXCERPT 6.8
• Point Estimates Not Labeled as Such
[We examined] the relationships between life satisfaction and two types of openness:
tolerance in relation to gays and lesbians and tolerance in relation to racial and ethnic minorities. Across all countries, there is a positive correlation between life satisfac-tion and both types of tolerance (.78, .63, gays and racial minorities, respectively).
Source: Florida, R., Mellander, C., & Rentfrow, P. J. (2010). Socioeconomic Structures and Happiness. Working Paper Series, Martin Prosperity Institute, Rotman School of Manage-ment, University of Toronto, REF. 2010–MPIWP-002.
EXCERPT 6.9
• Point Estimates Referred to as Point Estimates
Adult smoking prevalence for African Americans was 19.3% compared with 15.4%
for all Californians. The health care cost of smoking was $626 million for the African American community. Although African Americans account for 6% of the California adult population, they account for over 8% of smoking-attributable ex-penditures and fully 13% of smoking-attributable mortality costs. [However], our es-timates are point eses-timates and do not account for the sampling variability in smoking prevalence, relative risks, or health care expenditure estimates.
Source: Max, W., Sung, H., Tucker, L., & Stark, B. (2010). The disproportionate cost of smok-ing for African Americans in California. American Journal of Public Health, 100(1), 152–158.
in the excerpt’s first three sentences as fully accurate indices of smoking preva-lence, smoking-related health-care expenditures, or smoking-related mortality rates. Unfortunately, appropriate warnings such as this appear infrequently in research reports.
Point estimates are connected with many of the statistical summaries included in research reports: percentages, means, medians, standard deviations, to name but a few. Researchers also engage in point estimation in the discussion of the measur-ing instruments used to collect data. As was indicated in Chapter 4, these discus-sions often involve the presentation of reliability and validity coefficients.
Give yourself a pat on the back if you recall, from what I said in Chapter 4, that such coefficients are only estimates. If a different sample of examinees were to provide the data for the assessment of reliability or validity, the obtained coefficients most likely would fluctuate. Sampling error accounts for such fluctuation.
Although it is possible to build CIs around reliability and validity coeffi-cients, researchers rarely do this. Instead, point estimates are typically provided.
This is a common practice, even in cases where the researcher uses inferential statistical procedures in other parts of the research report. Consider, for exam-ple, Excerpts 6.10 and 6.11. The reliability coefficient in the first of these excerpts and the validity coefficient in second excerpt are point estimates, not parameters.
EXCERPTS 6.10–6.11
• Point Estimates of Reliability and Validity
Reliability analyses showed that the scale had high reliability in this study (alpha .93).
Source: Kwo, S. Y. C. L., & Shek, D. T. L. (2010). Personal and family correlates of suicidal ideation among chinese adolescents in Hong Kong. Social Indicators Research, 95(3), 407–419.
Concurrent validity coefficient between this scale and ENRICH Marital Satisfaction Questionnaire was 0.83.
Source: Rajabi, G. R. (2010). Factorial structure of marital satisfaction scale in married staff members of Shahid Chamran University. Iranian Journal of Psychiatry and Clinical Psychol-ogy, 15(4), 351–358.
Occasionally, you may come across a research report that exemplifies the good practice of building CIs around reliability and validity coefficients. Consider, for example, Excerpts 6.12 and 6.13. The researchers who conducted these studies deserve high praise for recognizing that their reliability and validity coefficients were sample statistics, not population parameters.
Although the likelihood of sampling error causes the practice of point esti-mation to seem quite ill-founded, this form of statistical inference deserves to be re-spected for two reasons. These two supportive arguments revolve around (1) the role played by point estimation in interval estimation and (2) the reliance on point estimation by more advanced scientific disciplines (such as physics). Let’s consider briefly each of these reasons why it is unwise to look on point estimation with com-plete disrespect.
When engaged in interval estimation, the researcher builds a CI that surrounds the sample statistic. Point estimation is relied on in two ways when such intervals are constructed. First, the pinpoint value of the sample statistic is used as the best single estimate of the population parameter. The desired interval is formed by adding a certain amount to the statistic and subtracting a certain amount from the statistic. Hence, the value of the statistic, as a point estimate of the parameter, serves as the foundation for each and every CI that is constructed.
Interval estimation draws on point estimation in a second manner. To be more specific, the amount that is added to (and subtracted from) the statistic in order to obtain the interval’s upper and lower limits is based on a point estimate of the pop-ulation’s variability. For example, when a CI is constructed around a sample mean, the distance between the end points of the interval is contingent on, among other things, a point estimate of the population standard deviation. Likewise, whenever a CI is built around a sample proportion, the length of the interval cannot be spec-ified until the researcher first uses point estimation to guess how variable the pop-ulation is.
EXCERPTS 6.12–6.13
• Confidence Intervals Built Around Reliability and Validity Coefficients
Intraclass correlation coefficients (ICCs) (95% confidence interval [CI]) for inter-rater reliability were .90 (.71–.97), .92 (.77–.97), and .85 (.64 –.95) for time, num-ber of steps, and smoothness, respectively.
Source: Hess, R. J., Brach, J. S., Piva, S. R., & VanSwearingen, J. M. (2010). Walking skill can be assessed in older adults: Validity of the Figure-of-8 Walk Test. Physical Therapy, 90(1), 89–99.
To measure concurrent validity [of the CCAM] with the COVS, the Pearson corre-lation (r) between scores on the CCAM and COVS was calculated. . . . The Pearson correlation coefficient between the CCAM total score and the COVS total score was very high (r .96, 95% CI .911.00) for the measure as a whole.
Source: Huijbregts, M. P. J., Teare, G. F., McCullough, C., Kay, T. M., Streiner, D., Wong, S. K. C., et al. (2009). Standardization of the Continuing Care Activity Measure: A multicen-ter study to assess reliability, validity, and ability to measure change. Physical Therapy, 89(6), 546–555.
From a totally different perspective, the practice of point estimation deserves to be respected. Certain well-respected scientists assert that as a discipline advances and becomes more scientifically rigorous, point estimation is turned to with both increased frequency and greater justification.