Ox and Dog compatibility
Ox (Chou, 丑) and Dog (Xu, 戌): a punishment (Xing, 刑).
These are fixed relations among the twelve branches of the Chinese calendar, set out in the classic San Ming Tong Hui (三命通会), written in 1578, quoted below. They describe the signs, not two people: no classic gives a method for matching two people by their signs, so there is no score here.
What the book says
Punishment (Xing, 刑): the punishment of trading on power
Ox and Dog are two of the three signs, Ox, Dog and Goat, in the punishment of trading on power: each uses its strength to punish what the next one holds.
The Goat trades on the power of its Yin Fire to punish the Metal inside the Ox; the Ox trades on the power of its strong Water to punish the Fire inside the Dog; the Dog trades on the power of its Yin Metal to punish the Wood inside the Goat. So it is called the punishment of trading on power.
未恃丁火之势以刑丑中之金，丑恃旺水之势以刑戌中之火，戌恃辛金之势以刑未中之木，故曰恃势。
Seeing a punishment, do not at once call it ill. Look whether the five elements hold a good hour, strength, the officer star, the seal, a noble spirit, virtue or blessing; with these good things to help, the punishment does no harm and is put to use instead.
凡见刑不可便以凶论，须看五行中有无吉辰、旺相、官星、印绶、责神、德福等物，有此诸吉相扶相助，刑不为害，而反为用
San Ming Tong Hui, volume 2: On the three punishments (论三刑) · source text
Their elements
The Ox is Earth and the Dog is Earth: the same element.
How much it counts
A sign is one character of the eight in a BaZi chart, the year’s branch. The book reads these relations between the pillars of one chart, and tells the reader to look at the whole chart’s elements first:
Clash and break, three harmony and six harmony: where a chart has them, ask next how its five elements get on. Misfortune may give rise to blessing, and blessing to misfortune.
相冲相破，三合六合，命中有之，即求五行相得何如。或祸中生福，福中生祸
San Ming Tong Hui, volume 3: On the symbolic stars as a whole (总论诸神煞) · source text
Birth years, and 2027
- Ox: 1925, 1937, 1949, 1961, 1973, 1985, 1997, 2009, 2021. In 2027, the Year of the Fire Goat (Ding Wei, 丁未), the Ox clashes with the year (Chong Tai Sui, 冲太岁) and is punished by the year (Xing Tai Sui, 刑太岁).
- Dog: 1934, 1946, 1958, 1970, 1982, 1994, 2006, 2018. In 2027, the Year of the Fire Goat (Ding Wei, 丁未), the Dog is punished by the year (Xing Tai Sui, 刑太岁) and breaks with the year (Po Tai Sui, 破太岁).
A sign’s year is usually counted from Chinese New Year, between late January and mid-February; BaZi counts it from the Start of Spring, about 4 February. Born in January or early February? You may be the sign before: the sign pages check your date.
Two real people?
Each person’s chart has eight characters, and the same relations run between all of them, not only the two signs. The two-chart page sets both charts side by side, branch by branch, and gives what each chart says about marriage on its own. No score there either.
Ox with the other signs: Rat · Ox · Tiger · Rabbit · Dragon · Snake · Horse · Goat · Monkey · Rooster · Pig.
Dog with the other signs: Rat · Tiger · Rabbit · Dragon · Snake · Horse · Goat · Monkey · Rooster · Dog · Pig.