Two-tailed p-values

Figure 2. Decreased scar size in cell transplanted animals. The resnlts of the myocardial scar measurements are shown as means and corresponding 95% confidence intervals. Two tailed p values were calculated for equal variances and considered significant when p<0.05.

Denotes a two-tailed p value <. 05 vs. healthy brain, considered significant, denotes a p value >. 05 vs. healthy brain, considered not significant. 

Two multivariate analysis of the relationships between 5 biomechanical factors A, IPA, BQ, BM, H and the dependent variable CR were done using Classification Tree (CART, Systat.ll) and a Multinomial Logistic regression model (Equation. 1, Systat.ll). Two tailed p-values of 0.05 or less was used to evaluate statistical significance of predictor variables (a, b, c, d, e and constant in Equation 1). 

Fisher s probabilities are not necessary symmetric. Although some analysts will double the one-tailed p value to obtain the two-tailed result, this method is usually overly conservative. 

An unnecessary complication, which was possibly once introduced to make life easier, is the distinction between one- and two-tailed Student /-values (tails are also used in other statistics). Two-tailed probabilities are spread over the two ends of the distribution with half the given probability in each tail, and are denoted by putting a double prime (") after the probability value. One-tailed probabilities are shown as a single prime ( ) and refer to just one tail of the distribution. For example, for a 95% confidence interval and 10 degrees of freedom, 0.025, 10 is equal to o.o5",io> as can be seen from figure 2.9. Annoyingly, in Excel the z values obtained from the normal distribution are always one tailed (=—NORMSINV(p)1) but the Student /-values... 

Data from ref. 34 Data from ref. 35 Data from tef.36 Probability values (p) of the occuirence of abnormal litters or fetuses by comparison of treatment with alkaloid vs. earner controls obtained by chi-square analysis using two-tailed Fisher exact results One-tailed p-values, students t vs. controls. 

We used a two-tailed test. Upon rereading the problem, we realize that this was pure FeO whose iron content was 77.60% so that p = 77.60 and the confidence interval does not include the known value. Since the FeO was a standard, a one-tailed test should have been used since only random values would be expected to exceed 77.60%. Now the Student t value of 2.13 (for —to05) should have been used, and now the confidence interval becomes 77.11 0.23. A systematic error is presumed to exist. 

Let us digress a moment and consider when a two-tailed test is needed, and what a one-tailed test implies. We assume that the measurements can be described by the curve shown in Fig. 2.10. If so, then 95% of the time a sample from the specified population will fall within the indicated range and 5% of the time it will fall outside 2.5% of the time it is outside on the high side of the range, and 2.5% of the time it is below the low side of the range. Our assumption implies that if p does not equal the hypothesized value, the probability of its being above the hypothesized value is equal to the probability of its being below the hypothesized value. 

There will be incidences when the foregoing assumptions for a two-tailed test will not be true. Perhaps some physical situation prevents p from ever being less than the hypothesized value it can only be equal or greater. No results would ever fall below the low end of the confidence interval only the upper end of the distribution is operative. Now random samples will exceed the upper bound only 2.5% of the time, not the 5% specified in two-tail testing. Thus, where the possible values are restricted, what was supposed to be a hypothesis test at the 95% confidence level is actually being performed at a 97.5% confidence level. Stated in another way, 95% of the population data lie within the interval below p + 1.65cr and 5% lie above. Of course, the opposite situation might also occur and only the lower end of the distribution is operative. 

The P resulting from these computations will be the exact one- or two-tailed probability depending on which of these two approaches is being employed. This value tells us if the groups differ significantly (with a probability less than 0.05, say) and the degree of significance. 

Fig. 4.7 (a) Staphylococcus aureus or Escherichia coli (108 cells per mL) was incubated for lOmin with BF1-BF3 at 100pM concentration in PBS followed by illumination with 400-700 -nm light at an irradiance of 200mW/cm2. Aliquots were removed from the suspensions after the various fluences of light had been delivered and CFU determined. Values are means of six independent experiments and bars are SEM. P < 0.05, P < 0.01 two-tailed unpaired /-test, (b) S. aureus was incubated with 1 pM concentration of BF4-BF6 and E. coli (both 108 cells per mL) was incubated with BF4-BF6 at lOpM concentrations for 10 min followed by a wash and illumination with white light. Values are means of six independent experiments and bars are SEM. P < 0.05, P < 0.01, P < 0.001 two-tailed unpaired /-test... 

The p-value calculation detailed in the previous section gives what we call a two-tailed or a two-sided test since we calculate p by taking into account values of the test statistic equal to, or more extreme, than that observed, in both directions. So... 

The statistical test procedures that we use unfortunately are not perfect and from time to time we will be fooled by the data and draw incorrect conclusions. For example, we know that 17 heads and 3 tails can (and will) occur with 20 flips of a fair coin (the probability from Chapter 3 is 0.0011) however, that outcome would give a significant p-value and we would conclude incorrectly that the coin was not fair. Conversely we could construct a coin that was biased 60 per cent/40 per cent in favour of heads and in 20 flips see say 13 heads and 7 tails. That outcome would lead to a non-significant p-value (p = 0.224) and we would fail to pick up the bias. These two potential mistakes are termed type I and type II errors. 

Critical values with a = 0.05 (P = 95%), two-tailed values for /- -tests (see also next table)... 

Note. The critical values of f are appropriate for a two-tailed test. For a one-tailed test, use the f value from the column with twice the P value. 

The experiments described here are principally diagnostic in nature where cellular biomass was significantly enhanced in bottles after resource (iron or light) amendment, relative to control (or other) treatments, we infer that algal growth rates in the control (or other) treatments were limited by a deficiency in that resource. The statistical significance of differences between mean values of parameters measured in different treatments were assessed using a two-tailed r-test for comparisons between two treatments, or a one-way analysis of variance (ANOVA) for comparisons between three or more treatments, at a confidence level of 95% (P = 0.05). 

Data are expressed as the means SE. Statistical significance is assessed by two-tailed unpaired Student s t-test or one way analysis of variance (ANOVA) followed by either Dunnett s test for multiple comparisons vs. control or the Newman-Keuls test for all pair-wise comparisons. Tests indicating a value of P < 0.05 indicate a statistically significant difference between groups. 

Zinc and copper values are expressed as mean S.D. Zinc and copper were measured using flame atomic absorption spectrophotometry. A two tailed t-test was used to compare the means for copper (p = 0.8) and for zinc (p = 0.6). 

The statistical parameters generated in the process of fitting the data to the equation are also used to determine the significance of the equation. A common criterion is to retain coefficients if their two-tailed probability is less than 0.05 P(2-tail) < 0.05. A two-tailed probability smaller than 0.05 means that the deviation from the true value lies in the positive or negative regions of the normal error curve corresponding to less than 5% of the area. It... 

Table 41.2 Critical values of Student s t statistic (for two-tailed tests). Reject the null hypothesis at probability P if your calculated t value exceeds the value shown for the appropriate degrees of freedom = (ni — 1) + (02 1)...

On many occasions, sample statistics are used to provide an estimate of the population parameters. It is extremely useful to indicate the reliability of such estimates. This can be done by putting a confidence limit on the sample statistic. The most common application is to place confidence limits on the mean of a sample from a normally distributed population. This is done by working out the limits as F— ( />[ i] x SE) and F-I- (rr>[ - ij x SE) where //>[ ij is the tabulated critical value of Student s t statistic for a two-tailed test with n — 1 degrees of freedom and SE is the standard error of the mean (p. 268). A 95% confidence limit (i.e. P = 0.05) tells you that on average, 95 times out of 100, this limit will contain the population... 

---