Question

Biologists studied the relationship between embryonic heart rate and egg mass for 20 species of birds. They found that heart rate, \(Y\), has a linear relationship with the logarithm of egg mass, \(X .\) The data are given in the following table. \({ }^{37}\) For these data the fitted regression equation is \(\hat{y}=368.06-82.452 x\) and \(\mathrm{SS}(\) resid \()=15748.6\) (a) Interpret the value of the intercept of the regression line, \(b_{0}\), in the context of this setting. (b) Interpret the value of the slope of the regression line, \(b_{1}\), in the context of this setting. (c) Calculate \(s_{e}\) and specify the units. (d) Interpret the value of \(s_{e}\) in the context of this setting.

Short answer

The intercept (368.06) represents the expected embryonic heart rate for an egg mass of 1 gram. The slope (-82.452) indicates the expected decrease in heart rate per unit increase in the log of egg mass. The standard error \(s_e \approx 29.85\) beats per minute measures the typical deviation of the observed heart rates from the regression line.

Step-by-step solution

Interpreting the intercept

The intercept, denoted by \(b_0\), is the value of \(Y\) when \(X\) is zero. In the context of this setting, it represents the expected embryonic heart rate (in beats per minute) for a bird species with an egg whose mass has a logarithm of zero (i.e., an egg mass of 1 gram, since \(\log(1) = 0\)). The value of the intercept is \(368.06\).

Interpreting the slope

The slope, symbolized by \(b_1\), describes the change in the response variable \(Y\) for each one-unit change in the predictor variable \(X\). Here, the slope of \(-82.452\) indicates that for each increase of 1 in the logarithm of egg mass, the expected embryonic heart rate decreases by \(82.452\) beats per minute.

Calculating the standard error of the estimate

The standard error of the estimate, \(s_e\), measures the typical distance that the observed values fall from the regression line. It is calculated as the square root of the residual sum of squares (SS(resid)) divided by the degrees of freedom, which is the number of observations \(n\) minus 2 (because there are two parameters estimated, the intercept and the slope). The residual sum of squares is \(15748.6\) and there are 20 bird species, so we calculate \(s_e = \sqrt{\frac{15748.6}{20 - 2}}\).

Determining the units of the standard error

The variable \(Y\) represents the embryonic heart rate in beats per minute. Hence, the units of the standard error, which measures the deviations of the observed \(Y\) from the fitted values, will be in the same units as \(Y\), that is, beats per minute.

Interpreting the standard error

The standard error \(s_e\) quantifies the typical error of the estimated embryonic heart rate from the actual heart rate. It provides an indication of the accuracy of the predicted heart rate from the regression model in beats per minute.

Key concepts

Linear Regression

Linear regression is a fundamental statistical method used to model the relationship between a dependent variable, often denoted as Y, and one or more independent variables, known as X. In the context of studying embryonic heart rate and egg mass in birds, researchers used linear regression to examine how changes in the logarithm of egg mass (the independent variable X) affected the embryonic heart rate (the dependent variable Y).

The equation generated from linear regression takes the form \(\hat{y}=b_0 + b_1x\), where \(\hat{y}\) is the predicted value of Y, \(b_0\) is the y-intercept of the regression line, and \(b_1\) is the slope. This equation helps predict the embryonic heart rate based on a given egg mass. Understanding the relationship between these two variables can offer insights into the biological processes affecting bird species during development.

Standard Error

The standard error (SE) is an important measure in regression analysis as it reflects the average distance that the observed data points deviate from the regression line. It essentially gives us an idea of the precision of the regression estimate. Calculated from the residual sum of squares, the standard error provides insight into the variability of the data. In the field of biostatistics, a lower standard error indicates a more precise estimate of the relationship between variables. For the study of embryonic heart rate, the standard error tells us how much the observed heart rates typically vary from the heart rates predicted by the model.

Residual Sum of Squares

The residual sum of squares (SS(resid)) is a measure used to quantify the discrepancy between the data and an estimation model—a lower SS(resid) suggests a better fit. It can be thought of as the sum of the squares of the vertical differences between the observed values and the values predicted by the regression line. In mathematical terms, it is calculated as \(SS(\text{resid}) = \sum (y_i - \hat{y}_i)^2\), where \(y_i\) is the observed value and \(\hat{y}_i\) the predicted value. In the context of the bird study, a residual sum of squares of 15748.6 indicates the level of variation in embryonic heart rate that is not explained by the logarithm of egg mass alone.

Biostatistics

Biostatistics is the application of statistics to biological fields such as medicine, biology, and public health. It involves the design of biological experiments, the collection and analysis of data from those experiments, and the interpretation of the results. Biostatisticians use regression analysis to study relationships between variables in various biological contexts. In the case of the embryonic heart rate and egg mass study, biostatistics plays a critical role in understanding these biological variables' patterns, thereby facilitating knowledge of avian development.

Embryonic Heart Rate

Embryonic heart rate refers to the frequency of the heartbeat in the developing embryo. It is a critical indicator of developmental health and vitality. In many species, including birds, the embryonic heart rate can be influenced by various factors, such as genetic characteristics and environmental conditions. The biologists' research in the given exercise aims to explore how the embryonic heart rate in bird species is influenced by the mass of the egg, as understanding this relationship may provide valuable insights into species-specific developmental processes and energy demands during incubation.