## Introduction

In the previous article (Scott and others 2011), we focussed on exploration and the more subjective – but still important – aspects of visualising and summarising data, using material from a study by Bell and others (2011) (Table 1). In this article, we will highlight some more formal statistical methods, namely, hypothesis tests and confidence intervals, and introduce them for two simple experimental situations. Underpinning these methods is the idea of a probability model, and, in the last article, we introduced the normal or Gaussian distribution and some of its properties, which will serve for our model. We also showed an example of a probability plot to verify the particular distributional assumption we make. To highlight the clinical use of some of these statistical methods, we will introduce a further study in this article, an investigation of the anaesthetic sparing effect of brachial plexus block (BPB) in cats (Mosing and others 2010; Table 1). Although various quantities were measured in their experiment, we will consider only the cardiorespiratory variables (heart rate, respiratory rate and non-invasive arterial blood pressure), and, for simplicity, we will simply look at one timepoint, although measurements were made at five timepoints (including all five timepoints in our analysis will become the topic of a later article).

Bell and others (2011) performed a blinded investigation assessing the sedative, cardiorespiratory and propofol-sparing effects in dogs of two doses of dexmedetomidine in combination with buprenorphine, compared with acepromazine in combination with buprenorphine. Sixty dogs (20 in each group) were recruited to the study, although 1 dog was subsequently excluded. Heart rate, arterial blood pressure, respiratory rate and quality of sedation were recorded by the authors, as well as propofol dose requirements. |

Mosing and others (2010) performed a blinded study evaluating the isoflurane sparing effect and the postsurgical analgesia provided by a brachial plexus nerve block in cats undergoing distal thoracic limb surgery. Twenty cats were recruited to the study, with 10 undergoing conventional anaesthesia and another 10 undergoing anaesthesia combined with a BPB. Two cats were subsequently excluded from the study (one from each group). The investigators recorded a number of cardiovascular and respiratory variables, in addition to isoflurane requirements and postoperative pain scores. |

Hypothesis testing and construction of confidence intervals are part of the process of statistical inference, and Fig 1 tries to explain what inference is about.

Statistical inference is the process whereby, on the basis of the experiment we performed, the data observed and a statistical model, we draw conclusions and make statements about the real world, not restricted to the experiment we carried out.

Before proceeding to introduce hypothesis testing and confidence intervals, it is necessary to define a basic vocabulary that is commonly used.

A parameter is generally unknown (usually because it is not possible to measure the variable on the entire population); the corresponding statistic and the variability of that statistic are then used as a tool to make inferences about the unknown parameter. This means we make a statement about a specific sample and then try to generalise it to the entire population.

We should be particularly concerned about how the sample is to be extracted from the population and how representative of the population the sample actually is that we have collected. We have seen in the previous article (Scott and others 2011) how data from the paper of Bell and others (2011) demonstrated that heart rate varied amongst study subjects even before they were assigned to any treatment, so that handling variability is an important aspect of any statistical analysis. If we repeated the experiment by Bell and others (2011), and so identified a further sample of 60 subjects, they also would vary in terms of heart rate; this is sometimes referred to as *sampling variability*.

The objectives of our designed experiment are concerned not with the properties of just the individuals that we have been working with, but with the properties of the wider population. In specifying the objectives, we might use statements such as “is the effect of drug A on blood pressure significant?”, “is there a difference between the mean heart rate of dogs on treatment A compared with treatment B?” and so on. The analysis of the results from suitably designed experiments will be used to answer such questions, but we only have a sample from the population and we cannot, therefore, be 100% certain about the true value of the population attribute (e.g. arterial blood pressure). The results of our experiment are subject to uncertainty, and in seeking to answer the experimental question, we need to take account of both sampling variability and also any intrinsic uncertainty (e.g. if the equipment used was only calibrated to measure arterial blood pressure to within the nearest 10 mmHg, there will be an inherent inaccuracy in the answer we obtain). There is a variety of different hypothesis tests and confidence intervals depending on the experimental design and the questions of interest, but all have certain principles in common. This article will first describe those principles and then take two simple examples to explain those principles in action.