An Illustration of Obtaining p-values and Confidence Intervals Through the Use of The Significance Function

In this paper, we define the significance function of a scalar parameter of interest based on a statistical model. We demonstrate how confidence intervals and p-values can easily be obtained from the significance function directly. The focus of this paper is on the application of the significance function rather than technical details.


Introduction
Statistical inference is a topic that is covered in most, if not all, introductory statistics courses. This topic is further separated into two major concepts: confidence intervals and significance testing. In general, they are treated as two completely different concepts with different sets of required computations. As an illustration, let  X σ Note that, in practice, many models (e.g., the curve exponential model where the dimension of the parameter is greater than the dimension of the sufficient statistic) cannot be reduced in this form. In those situations, a common approach is to apply the likelihood based asymptotic method to obtain approximate inference about . of producing an interval containing the true value of the parameter, as seen in Moore et al. [2]. Nowadays, a reported confidence interval is generally defined as a two-sided interval with equal tail probabilities. However, one can modify it to be a one-sided interval, or with non-equal tail probabilities. As an illustrative example, Susic [3] studied the blood phosphorus level in dialysis patients and the data for one patient (in milligrams of phosphorus per deciliter of blood) are: 5.4 5.2 4.5 4.9 5.7 6.3 Assuming the data are independent and from a normal distribution, the t.test command in R generates a 95% confidence interval for the mean blood phosphorus level of (4.67, 5.99). As such, in general, over repeated sampling, 95% of similarly constructed confidence intervals as (4.67, 5.99) are expected to contain the population mean. Note that there are books and practitioners that will interpret the confidence interval as the probability that will fall within the obtained interval and use it for decision-making. This is not exactly how Neyman [1] originally defined confidence intervals. However, as discussed in Fraser [4], there are special cases that the two interpretations are the same.
In contrast, Fisher [5] defined the p-value as the probability, assuming that the null hypothesis is true, that the test statistic will take a value at least as extreme as the actually observed, as seen in Moore et al. [2]. In fact, Fisher's idea was not to use the p-value for decision-making [5]. Rather, if there is a p-value of 8%, this means that exactly 8% of potential data values are at least as extreme as the observed data, relative to the distribution with the parameter 0 . θ θ = Continuing with our example, to test the hypothesis the t.test command in R yields a p-value of 0.046. Thus, 4.6% of potential data values are to the right of the observed data and 95.4% of potential data values are to the left of the observed data, relative to the distribution with the parameter 4.8.

µ =
However, contrary to common practice, this is not intended for decision-making [4].
The rest of the paper is organized as follows. In Section 2, we introduce the significance function and how confidence intervals and p-values can be directly obtained from this function. In Section 3, examples are given to illustrate applications of the significance function. Some concluding remarks are given in Section 4.

Significance function
Fraser [6] briefly discussed the p-value function, also known as the significance function. In this paper, we officially define this function and illustrate how it can be used to obtain statistical inference. Let ( ) 1 , , n x x  be a sample of size n from a known statistical model and ψ be the scalar parameter of interest.

Moreover, let ( )
T ψ be the test statistic and ( ) t ψ be the corresponding observed values of the test statistic. Then, the significance function ( ) p ψ is defined as the probability that ( ) T ψ is less than or equal to the observed ( ) t ψ for a given ψ value. That is: is the cumulative distribution function of the test statistic for a given ψ value. Otherwise, ( ) p ψ is the survivor function of the test statistic for a given ψ value. Figure 1 is the significance function of the normal mean for the blood phosphorus level example discussed in Section 1.
To continue with our blood phosphorus level example from the previous section, the confidence interval is recorded in Figure   1. Since ( )

Applications of the significance function
In this section, we use two examples to illustrate the application of the significance function. [7] reported the survival times of male mice exposed to gamma radiation in weeks. The data are: θ Figure 2 gives the significance function   Example 2: Kieler et al. [8] reported that the pregnancy length in singleton pregnancies with a spontaneous onset labour, calculated from bipariental diameter of 509 mothers who are smokers, had a mean of 280.3 days and a standard deviation 8.2 days. Assume the length of these types of pregnancy is normally distributed. The parameter of interest is the precision parameter , ψ which is the reciprocal of the variance parameter. Since, which is given in Figure 3. As indicated in Figure 3

Conclusion
In this paper, we defined the significance function and demonstrated how it can be applied. Our illustrative examples showed that we can obtain both confidence intervals and p-values with ease directly from the significance function. Throughout this paper, we assumed that the significance function exists explicitly. However, in practice, this may not be the case. In these situations, there are standard approximations that can be used, such as the likelihood based methods (Wald method, Rao method, and the Wilks method), to obtain an approximate significance function. Applications of the likelihood based methods can be found in many books, such as Lawless [9], Kalflesich & Prentice [10], and Brazzele et al. [11].