Results
Table 1.
This table shows that, because all of the major variables (Treatment, Dose, Position) and their interactions have p-values less than 0.05, they all significantly affect the outcome being examined. The strength of these effects is revealed by the F-values.
Table 1.
This table shows that, because all of the major variables (Treatment, Dose, Position) and their interactions have p-values less than 0.05, they all significantly affect the outcome being examined. The strength of these effects is revealed by the F-values.
| Source of variation |
Degrees of freedom |
Sum of squares |
Mean squares |
F-value |
P-value |
| Treatment |
6 |
0.84690 |
0.14115 |
10.54** |
0.0000 |
| Dose |
1 |
0.11905 |
0.11905 |
8.89** |
0.0034 |
| Position |
4 |
4.29733 |
1.07433 |
80.21** |
0.0000 |
| Treatment*Dose |
6 |
0.34412 |
0.05735 |
4.28** |
0.0005 |
| Treatment*Position |
24 |
1.15051 |
0.04794 |
3.58** |
0.0000 |
| Dose*Position |
4 |
0.22856 |
0.05714 |
4.27** |
0.0027 |
| Treatment*Dose*Position |
24 |
1.09450 |
0.04560 |
3.40** |
0.0000 |
| Error |
140 |
1.87527 |
0.01339 |
|
|
| Total |
209 |
9.95624 |
|
|
|
Table 2.
Comparison of means; (A) Dose x Treatment interaction mean±SE.
Table 2.
Comparison of means; (A) Dose x Treatment interaction mean±SE.
| Dose |
|
|
|
Treatment |
|
|
|
Mean |
| T1 |
T2 |
T3 |
T4 |
T5 |
T6 |
T7 |
| Dose 1 |
1.46±0.08abc |
1.41±0.03cd |
1.43±0.06bc |
1.42±0.03bc |
1.44±0.03bc |
1.35±0.04cd |
1.37±0.11cd |
1.41±0.02B |
| Dose 2 |
1.48±0.05abc |
1.55±0.06ab |
1.47±0.05abc |
1.59±0.06a |
1.48±0.02abc |
1.38±0.03cd |
1.27±0.04d |
1.46±0.02A |
| Mean |
1.47±0.05a |
1.48±0.04a |
1.45±0.04ab |
1.51±0.04a |
1.46±0.02a |
1.36±0.02bc |
1.32±0.06c |
|
The effects of different dosages (Dose 1 and Dose 2) were examined across several treatments (T1 to T7), and the results are summarized in the table below. The average response to each dose across all treatments is shown in the “Mean” row. For each treatment-dose combination, the values in the table show the means along with the corresponding standard deviations (marked by). For statistically significant differences between treatments within the same dose group, the letters (a, b, c, d, A, B) are employed as indicators. Treatments that share the same letter do not differ greatly from one another. As an illustration, the means of treatments T2 and T4 in Dose 1 are similar (1.410.03cd and 1.420.03bc, respectively), as they both share the same letters “c” and “d,” indicating no discernible difference between them.
In conclusion, the table offers a clear way to contrast the results of several treatments at two distinct doses, with letters designating which treatments are statistically equivalent or dissimilar.
Table 3.
(B) Position x Treatment interaction Mean ±SE.
Table 3.
(B) Position x Treatment interaction Mean ±SE.
| Dose |
|
|
|
Treatment |
|
|
|
Mean |
| |
T1 |
T2 |
T3 |
T4 |
T5 |
T6 |
T7 |
|
| P1 |
1.41±0.02d-g |
1.48±0.02de |
1.45±0.03d-g |
1.46±0.04def |
1.50±0.03de |
1.31±0.04e-h |
1.21±0.02fgh |
1.40±0.02BC |
| P2 |
1.35±0.02e-h |
1.40±0.03d-h |
1.35±0.05e-h |
1.42±0.02d-g |
1.44±0.03d-g |
1.33±0.02e-h |
1.15±0.04h |
1.35±0.02CD |
| P3 |
1.30±0.03e-h |
1.31±0.02e-h |
1.31±0.03e-h |
1.34±0.03e-h |
1.33±0.03e-h |
1.29±0.02e-h |
1.20±0.01gh |
1.30±0.01D |
| P4 |
1.38±0.03e-h |
1.44±0.03d-g |
1.50±0.04de |
1.53±0.02cde |
1.50±0.02de |
1.39±0.02e-h |
1.22±0.03fgh |
1.42±0.02B |
| P5 |
1.90±0.10a |
1.76±0.11abc |
1.65±0.14a-d |
1.79±0.13ab |
1.54±0.04b-e |
1.51±0.10de |
1.81±0.17a |
1.71±0.05A |
This table displays results from an experiment in which seven treatments (T1–T7) were used to examine the effects of different dosages (Dose) given to five different groups or populations (P1–P5). The response average for each population across all treatments is shown in the “Mean” row. The values in the table represent the average outcomes for each treatment-population combination, along with the corresponding standard deviations (marked by). The following letters are used to denote statistically significant differences between treatments within the same population group: a, b, c, d, e, f, g, h, A, B, C, D. The differences between treatments beginning with the same letter are not very distinct. For instance, treatments T1, T3, and T6 in Population 1 (P1) had comparable averages (1.410.02d-g, 1.450.03d-g, and 1.310.04e-h), as they share the same letters ‘d’, ‘e’, ‘f’, ‘g’, and ‘h,’ showing no substantial difference between them. The aggregate average of each population’s responses is displayed in the “Mean” row. The mean response, for Population 1, is 1.40 0.02 BC for all treatments.
Table 4.
(C) Dose x Position x Treatment interaction Mean ±SE.
Table 4.
(C) Dose x Position x Treatment interaction Mean ±SE.
| Dose |
Position |
|
|
|
Treatment |
|
|
|
Mean |
| |
|
T1 |
T2 |
T3 |
T4 |
T5 |
T6 |
T7 |
|
| Dose 1 |
P1 |
1.36±0.01f-j |
1.44±0.04d-j |
1.38±0.03f-j |
1.39±0.04f-j |
1.45±0.01d-j |
1.28±0.04f-j |
1.17±0.04ij |
1.35±0.02BCD |
| |
P2 |
1.31±0.01f-j |
1.34±0.02f-j |
1.32±0.05f-j |
1.39±0.02f-j |
1.38±0.02f-j |
1.30±0.00f-j |
1.10±0.08j |
1.31±0.02CD |
| |
P3 |
1.28±0.05f-j |
1.28±0.02f-j |
1.24±0.03f-j |
1.28±0.03f-j |
1.26±0.01f-j |
1.26±0.01f-j |
1.19±0.02hij |
1.26±0.01D |
| |
P4 |
1.34±0.04f-j |
1.39±0.01f-j |
1.42±0.02e-j |
1.52±0.02d-i |
1.48±0.03d-j |
1.36±0.01f-j |
1.24±0.07f-j |
1.39±0.02BC |
| |
P5 |
2.00±0.18ab |
1.59±0.06c-g |
1.81±0.11a-d |
1.54±0.12d-i |
1.63±0.04b-f |
1.54±0.14c-i |
2.13±0.16a |
1.75±0.06A |
| Dose 2 |
P1 |
1.46±0.01d-j |
1.52±0.01d-i |
1.52±0.01d-i |
1.53±0.02d-i |
1.54±0.03c-i |
1.34±0.06f-j |
1.24±0.01f-j |
1.45±0.03B |
| |
P2 |
1.38±0.01f-j |
1.47±0.02d-j |
1.39±0.10f-j |
1.45±0.01d-j |
1.50±0.05d-i |
1.36±0.01f-j |
1.20±0.01g-j |
1.39±0.02BC |
| |
P3 |
1.32±0.01f-j |
1.33±0.04f-j |
1.37±0.03f-j |
1.41±0.03f-j |
1.40±0.01f-j |
1.31±0.02f-j |
1.21±0.02g-j |
1.34±0.02CD |
| |
P4 |
1.42±0.04e-j |
1.50±0.00d-i |
1.57±0.04c-h |
1.53±0.02d-i |
1.51±0.03d-i |
1.42±0.01e-j |
1.19±0.01hij |
1.45±0.03B |
| |
P5 |
1.81±0.10a-e |
1.93±0.16abc |
1.49±0.25d-j |
2.03±0.07a |
1.45±0.02d-j |
1.47±0.17d-j |
1.49±0.12d-j |
1.67±0.07A |
The table displays the outcomes of an experiment in which five distinct positions each received a different dose of a treatment, and the effects were calculated across seven treatments. Which treatments are statistically comparable or different within each dose-position group are denoted by letters next to the mean values. Because they all share the same letters “d,” “f,” “g,” “h,” “i,” and “j,” treatments T2, T4, and T5 have similar means (1.440.04d-j) in Dose 1 and Position P1, showing no discernible difference between them. The “Mean” row displays the average reaction across all dose and location combinations. For instance, 1.350.02BCD is the mean response for Dose 1 and Position P1 across all treatments. In essence, the table offers a clear means of comparing the impacts of various treatments on five distinct positions for each of the two doses, with letters designating which treatments are statistically comparable or different within each dose-position group.
In conclusion, the table offers a clear method to assess the effects of various treatments on five distinct positions for each of the two dosages, with letters designating which treatments are statistically comparable or different within each dose-position group. Using concise language
Table 5.
Analysis of variance table for Head & Neck.
Table 5.
Analysis of variance table for Head & Neck.
| Source of variation |
Degrees of freedom |
Sum of squares |
Mean squares |
F-value |
P-value |
| Treatment |
6 |
1.85225 |
0.30871 |
26.66** |
0.0000 |
| Machine |
1 |
0.28229 |
0.28229 |
24.38** |
0.0000 |
| Position |
6 |
2.16754 |
0.36126 |
31.20** |
0.0000 |
| Treatment*Machine |
6 |
0.14921 |
0.02487 |
2.15* |
0.0497 |
| Treatment*Position |
36 |
0.43392 |
0.01205 |
1.04NS |
0.4144 |
| Machine*Position |
6 |
0.26335 |
0.04389 |
3.79** |
0.0014 |
| Treatment*Machine*Position |
36 |
0.24144 |
0.00671 |
0.58NS |
0.9734 |
| Error |
196 |
2.26933 |
0.01158 |
|
|
| Total |
293 |
7.65933 |
|
|
|
According to the table’s low P-values and high F-values, the “Treatment,” “Machine,” and “Position” components, as well as their interactions (aside from “Treatment Machine Position”), have a statistically significant effect on the study’s conclusion. Both the “Treatment Machine Position” interaction and the “Treatment*Position” interaction have P-values above 0.05, thus they are not statistically significant.
Table 7.
(A) Dose x Treatment interaction Mean ± SE.
Table 7.
(A) Dose x Treatment interaction Mean ± SE.
| Machine |
|
|
|
Treatment |
|
|
|
Mean |
| |
T1 |
T2 |
T3 |
T4 |
T5 |
T6 |
T7 |
|
Machine 1 |
1.38±0.03ab |
1.41±0.04a |
1.40±0.04a |
1.40±0.04a |
1.39±0.04ab |
1.35±0.02ab |
1.15±0.02d |
1.36±0.01A |
Machine 2 |
1.29±0.03bc |
1.35±0.03ab |
1.38±0.02ab |
1.37±0.03ab |
1.24±0.05cd |
1.29±0.02bc |
1.14±0.01d |
1.29±0.01B |
| Mean |
1.33±0.02ABC |
1.38±0.02ABC |
1.39±0.02A |
1.39±0.02AB |
1.32±0.03C |
1.32±0.02BC |
1.14±0.01D |
|
The performance of two machines (Machine 1 and Machine 2) throughout several treatments (T1 to T7) is shown in this table. The overall mean for each therapy is also shown in the table. The numbers in the table are measurements for each machine and treatment combination. Lowercase letters (a, b, c) are used to indicate statistically similar groups inside each machine, and uppercase letters (A, B, C, D) are used to indicate statistically similar groups across machines. The numbers are reported in the format “mean standard deviation.” For instance, Treatment T1 in Machine 1 has a mean of 1.38 and a standard deviation of 0.03, and it is statistically identical to Treatments T2, T3, T4, and T5 (denoted by the same letter “a”). The “Mean” column offers the average results for each procedure on both machines.
In conclusion, this table compares the performance of two machines (Machine 1 and Machine 2) under various treatments and highlights groupings of treatments that are statistically similar based on averages and standard deviations.
Table 8.
(B) Position x Treatment interaction Mean ± SE.
Table 8.
(B) Position x Treatment interaction Mean ± SE.
| Position |
|
|
|
Treatment |
|
|
|
Mean |
| |
T1 |
T2 |
T3 |
T4 |
T5 |
T6 |
T7 |
|
| P1 |
1.19±0.01 |
1.17±0.03 |
1.23±0.04 |
1.22±0.02 |
1.19±0.04 |
1.23±0.01 |
1.07±0.01 |
1.18±0.01D |
| P2 |
1.33±0.01 |
1.37±0.02 |
1.37±0.01 |
1.38±0.03 |
1.31±0.06 |
1.34±0.04 |
1.14±0.03 |
1.32±0.02B |
| P3 |
1.43±0.04 |
1.51±0.04 |
1.43±0.03 |
1.48±0.03 |
1.41±0.09 |
1.35±0.03 |
1.16±0.03 |
1.40±0.02A |
| P4 |
1.49±0.06 |
1.46±0.07 |
1.55±0.04 |
1.50±0.05 |
1.46±0.11 |
1.38±0.05 |
1.18±0.05 |
1.43±0.03A |
| P5 |
1.40±0.04 |
1.48±0.03 |
1.49±0.04 |
1.54±0.03 |
1.41±0.10 |
1.35±0.04 |
1.20±0.04 |
1.41±0.02A |
| P6 |
1.21±0.05 |
1.38±0.03 |
1.28±0.03 |
1.26±0.03 |
1.20±0.07 |
1.26±0.03 |
1.10±0.03 |
1.24±0.02CD |
| P7 |
1.29±0.05 |
1.30±0.08 |
1.39±0.05 |
1.34±0.05 |
1.24±0.07 |
1.33±0.05 |
1.16±0.03 |
1.29±0.02BC |
Data for various positions (P1 to P7) and their performance across various treatments (T1 to T7) are shown in the table below. The performance of a particular position under a certain treatment is represented in each table cell by a mean value with a standard deviation. Based on averages and standard deviations, uppercase letters (A, B, C, and D) are used to denote statistically related groups of positions. Positions P3, P4, P5, and P6 have statistically identical performance, which is denoted by the letters “A” for P3, P4, and P5, and “CD” for P6. The “Mean” row displays the overall means for each therapy, taking into account all positions.
Table 9.
(C) Dose x Position x Treatment interaction Mean ± SE.
Table 9.
(C) Dose x Position x Treatment interaction Mean ± SE.
| Dose |
Position |
|
|
|
Treatment |
|
|
|
Mean |
| |
|
T1 |
T2 |
T3 |
T4 |
T5 |
T6 |
T7 |
|
| M1 |
P1 |
1.17±0.02 |
1.14±0.03 |
1.16±0.02 |
1.19±0.02 |
1.18±0.02 |
1.23±0.01 |
1.07±0.03 |
1.16±0.01H |
| |
P2 |
1.33±0.02 |
1.38±0.01 |
1.37±0.02 |
1.35±0.05 |
1.35±0.03 |
1.39±0.07 |
1.11±0.06 |
1.33±0.02C-F |
| |
P3 |
1.50±0.04 |
1.54±0.09 |
1.47±0.06 |
1.53±0.04 |
1.51±0.05 |
1.34±0.05 |
1.14±0.06 |
1.43±0.04ABC |
| |
P4 |
1.59±0.07 |
1.47±0.15 |
1.56±0.08 |
1.54±0.08 |
1.58±0.05 |
1.42±0.09 |
1.16±0.10 |
1.47±0.04A |
| |
P5 |
1.41±0.05 |
1.50±0.07 |
1.55±0.05 |
1.56±0.06 |
1.54±0.07 |
1.39±0.07 |
1.24±0.07 |
1.46±0.03AB |
| |
P6 |
1.26±0.01 |
1.41±0.06 |
1.24±0.05 |
1.22±0.04 |
1.30±0.08 |
1.28±0.05 |
1.13±0.05 |
1.26±0.02E-H |
| |
P7 |
1.40±0.02 |
1.41±0.13 |
1.47±0.06 |
1.45±0.01 |
1.29±0.10 |
1.41±0.04 |
1.18±0.06 |
1.37±0.03A-E |
| M2 |
P1 |
1.20±0.00 |
1.20±0.05 |
1.29±0.05 |
1.24±0.04 |
1.20±0.10 |
1.24±0.01 |
1.06±0.01 |
1.20±0.02GH |
| |
P2 |
1.33±0.02 |
1.37±0.03 |
1.38±0.02 |
1.40±0.02 |
1.26±0.13 |
1.28±0.03 |
1.17±0.02 |
1.31±0.02D-G |
| |
P3 |
1.36±0.04 |
1.47±0.01 |
1.40±0.01 |
1.43±0.04 |
1.31±0.16 |
1.36±0.02 |
1.17±0.04 |
1.36±0.03B-E |
| |
P4 |
1.39±0.02 |
1.45±0.03 |
1.54±0.04 |
1.46±0.06 |
1.34±0.22 |
1.34±0.01 |
1.20±0.02 |
1.39±0.04A-D |
| |
P5 |
1.39±0.06 |
1.46±0.04 |
1.43±0.02 |
1.52±0.02 |
1.27±0.15 |
1.31±0.04 |
1.16±0.05 |
1.36±0.03B-E |
| |
P6 |
1.16±0.09 |
1.34±0.03 |
1.31±0.03 |
1.29±0.04 |
1.11±0.08 |
1.24±0.02 |
1.07±0.02 |
1.22±0.03FGH |
| |
P7 |
1.18±0.01 |
1.19±0.04 |
1.32±0.05 |
1.24±0.06 |
1.19±0.10 |
1.24±0.07 |
1.13±0.03 |
1.21±0.02GH |
Means sharing similar letters in a row or a column are statistically non-significant (P>0.05). Small letters represent comparisons among interaction means, and capital letters are used for the overall mean. The information in this table pertains to multiple placements (P1 to P7), two machines (M1 and M2), and a range of treatments (T1 to T7) and doses. Each cell’s numbers show the average performance along with its standard deviation. Positions with statistically similar performance are grouped using uppercase letters (A, B, C, D, etc.) based on their means and standard deviations. It enables comparison of the effects of various settings, tools, therapies, and dosages on performance. The “Mean” row displays the treatment-specific average values across all placements and dosages.
In conclusion, the table indicates statistically similar groups among positions and aids in the analysis of how various factors affect performance across positions, treatments, and doses.