Question

Some investigators think that the concentration \((\mathrm{mg} / \mathrm{mL})\) of a particular antigen in supernatant fluids could be related to onset of meningitis in infants. The accompanying data are typical of that given in plots in the article "Type-Specific Capsular Antigen Is Associated with Virulence in Late-Onset Group B Streptococcal Type III Disease" (Infection and Immunity [1984]: 124-129). Construct an ANOVA table, and use it to test the null hypothesis of no difference in mean antigen concentrations for the groups. $$\begin{array}{lllllll} \text { Asymptomatic infants } & 1.56 & 1.06 & 0.87 & 1.39 & 0.71 & 0.87 & \\ \text { Infants with late onset sepsis } & 1.51 & 1.78 & 1.45 & 1.13 & 1.87 & 1.89 & 1.071 .72 \\ \text { Infants with late onset meningitis } & 1.21 & 1.34 & 1.95 & 2.27 & 0.88 & 1.67 & 2.57 \end{array}$$

Short answer

The null hypothesis is tested using the calculated F statistic. A high F value typically means that the null hypothesis is rejected and there is a statistically significant difference in group means. However, the critical value for rejection of the null hypothesis depends on the significance level and degrees of freedom, which are taken from the F distribution table. If the calculated F statistic is larger than the critical value from the F distribution table, then the null hypothesis is rejected.

Step-by-step solution

Calculate group means

First, calculate the mean of antigen concentrations for each group. For Asymptomatic infants, the mean (M1) is \((1.56 + 1.06 + 0.87 + 1.39 + 0.71 + 0.87) /6 =1.076\). For Infants with late onset sepsis, the mean (M2) is \((1.51+1.78+1.45+1.13+1.87+1.89+1.07+1.72) / 8 = 1.552\). For Infants with late onset meningitis, the mean (M3) is \((1.21+1.34+1.95+2.27+0.88+1.67+2.57) / 7 = 1.698\)

Calculate grand mean

Next, calculate the grand mean, which is the mean of all the data points regardless of the group. The grand mean is \(\ G = (1.076*6 + 1.552*8 + 1.698*7) /(6+8+7) = 1.441\)

Calculating BG & WG sums

Now, calculate the Between groups sum of squares (SSB) which is the sum of squared differences between each group mean and the grand mean, weighted by the sample size of each group. SSB = \(6*(1.076-1.441)^2 + 8*(1.552-1.441)^2 + 7*(1.698 - 1.441)^2 = 0.996\). Likewise, calculate the Within groups sum of squares (SSW) which is the sum of squared differences within each group. SSW = \(\Σ_{i=1}^{3}\Σ_{j=1}^{n_{i}} (X_{ij} - M_{i})^2\) where \(X_{ij}\) is each value within a group and \(M_{i}\) is the mean of a group.

Calculating BG & WG mean squares

Calculate Between-group mean square (MSB), which is the between-group sum of squares divided by between-group degrees of freedom (dfB). dfB is number of groups minus 1 so dfB = 3-1 = 2. MSB = SSB/dfB. Likewise, calculate Within-group mean square (MSW), which is the within-group sum of squares divided by within-group degrees of freedom (dfW). dfW is total number of observations minus number of groups, so dfW = (6+8+7)-3 = 18. MSW = SSW/dfW.

Calculating the F statistic

Finally, calculate the F statistic. This is the ratio of the mean square between groups to the mean square within groups: F = MSB/MSW. This statistic is used to test the null hypothesis.

Key concepts

Statistical Analysis in Medical Research

Statistical analysis plays a crucial role in medical research, facilitating the understanding of data and helping researchers draw meaningful conclusions from studies. It involves various methods to summarize, visualize, and interpret data, thereby aiding the decision-making process.

In the context of medical studies, statistical analysis helps researchers to compare different groups and determine if observed differences are statistically significant or likely due to chance. For instance, comparing mean antigen concentrations across different patient groups is fundamental in establishing correlations with medical conditions such as meningitis in infants. An analysis of variance (ANOVA) is commonly employed to evaluate if there are any statistically significant differences between the means of three or more independent groups.

ANOVA accomplishes this by decomposing the data variability into 'Between-group' and 'Within-group' variations, allowing investigators to ascertain if the groups differ more than what would be expected by random variability alone. The method provides a systematic approach to testing hypotheses and is, therefore, an indispensable tool in the researcher's toolkit for medical studies.

Mean Antigen Concentrations

The concept of mean antigen concentrations is central to immunological studies where researchers measure the level of particular antigens in a biological sample. Antigens can trigger an immune response, and their concentrations can be indicative of a person's health status or risk for certain diseases.

In the given exercise, antigen concentrations are being analyzed across three groups of infants to identify a potential link with meningitis. These mean concentrations are calculated for each group, providing a single value that represents the average level of antigen present in the samples. This averaging facilitates a clearer comparison across groups, moving from individual data points to a broader overview of trends within the sampled populations.

It's important to ensure accuracy in calculating these means, as they form the basis for further statistical tests like ANOVA. Errors at this stage can have a cascading effect, leading to incorrect conclusions. Additionally, understanding the distribution and variation of these mean concentrations among the groups is fundamental to comprehending the underlying immunological dynamics and their possible clinical implications.

Null Hypothesis Testing

Null hypothesis testing is a statistical method used to determine whether there is enough evidence to reject a presumed position that there is no effect or no difference. In medical research, the null hypothesis often posits that there are no differences between groups or that a treatment has no effect.

In the case of the ANOVA used to analyze the antigen concentrations of infants, the null hypothesis states that there are no differences in the mean concentrations across the groups – asymptomatic infants, infants with late-onset sepsis, and infants with late-onset meningitis. By testing this hypothesis, researchers can determine if the observed differences in group means are statistically significant, and thus potentially related to the medical condition of interest.

The 'F statistic' obtained in an ANOVA is compared against a critical value from an F-distribution to decide if the null hypothesis can be rejected. A high F value indicates that the variability between group means is larger than expected under the null hypothesis. If this F value is higher than the critical value for a given significance level, the null hypothesis is rejected, suggesting the group means are significantly different.